Differential Conformal Superalgebras and their Forms

Victor Kac,^{1} Michael Lau,^{2∗} and Arturo Pianzola^{3,4}^{∗}

1M.I.T., Department of Mathematics, Cambridge, Massachusetts, USA 02139 Email: kac@math.mit.edu

2University of Windsor, Department of Mathematics and Statistics, Windsor, Ontario, Canada N9B 3P4 Email: mlau@uwindsor.ca

3University of Alberta, Department of Mathematical and Statistical Sciences, Edmonton, Alberta, Canada T6G 2G1 Email: a.pianzola@math.ualberta.ca

4Instituto Argentino de Matemática, Saavedra 15, (1083) Buenos Aires, Argentina

Abstract. We introduce the formalism of differential conformal superalgebras, which we show leads to the “correct” automorphism group functor and accompanying descent theory in the conformal setting. As an application, we classify forms of N = 2 and N = 4 conformal superalgebras by means of Galois cohomology.

Keywords: Differential conformal superalgebras, superconformal algebras, Galois cohomology, infinite-dimensional Lie algebras.

MSC: 17B69 (Primary); 17B65,12G05,17B81 (Secondary)

Contents

0 Introduction 2

1 Conformal superalgebras 7 1.1 Differential rings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 1.2 Differential conformal superalgebras . . . . . . . . . . . . . . . . . . 9 1.3 Base change . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 1.4 The automorphism functor of a conformal superalgebra . . . . . . . 13 2 Forms of conformal superalgebras and Čech cohomology 15

∗Both authors gratefully acknowledge the continuous support of the Natural Sciences and Engineering Research Council of Canada.

2.1 Forms split by an extension . . . . . . . . . . . . . . . . . . . . . . . 16 2.2 Cohomology and forms . . . . . . . . . . . . . . . . . . . . . . . . . . 19 2.3 Limits . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25 2.4 The centroid trick . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 2.5 Central extensions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31

3 Examples and applications 33 3.1 Current conformal superalgebras . . . . . . . . . . . . . . . . . . . . 35 3.2 The Virasoro conformal algebra . . . . . . . . . . . . . . . . . . . . . 39 3.3 Forms of the N = 2 conformal superalgebra . . . . . . . . . . . . . . 40 3.4 Forms of the N = 4 conformal superalgebra . . . . . . . . . . . . . . 46

0 Introduction

The families of superconformal algebras described in the work of A. Schwimmer and N. Seiberg [17] bear a striking resemblance to the loop realization of the affine Kac-Moody algebras [8]. All of these algebras belong to a more general class known as Γ-twisted formal distribution superalgebras, where Γ is a subgroup of ℂ containing ℤ [10, 11].

In a little more detail, a superconformal algebra^{1} (or more generally, any twisted formal distribution algebra), is encoded by a conformal superalgebra A and an automorphism σ : A → A. Recall that A has a ℂ[∂]-module structure and n-products a_{(n)}b, satisfying certain axioms [9]. Let σ be a diagonalizable automorphism of A with eigenspace decomposition

A = ⨁_{\overline{m}∈Γ/ℤ}A\overline{m},

where A\overline{m} = \{a ∈ A | σ(a) = e^{2πim}a\}, Γ is an additive subgroup of ℂ containing ℤ, and \overline{m} ∈ ℂ/ℤ is the coset m+ℤ ⊂ ℂ.^{2} Then the associated Γtwisted formal distribution superalgebra Alg (A, σ) is constructed as follows.

1e.g., the Virasoro algebra or its superanalogues

2If σ^{M} = 1, then this construction can be performed over an arbitrary algebraically closed field k of characteristic zero in the obvious way, by letting Γ be the group \frac{1}{M} ℤ and replacing e^{2πi/M} with a primitive Mth root of 1 in k. This is the situation that will be considered in the present work.

Let L(A, σ) = ⨁_{m∈Γ} (A\overline{m} ⊗_{ℂ} t^{m}) , and let

Alg (A, σ) = L(A, σ) / (∂ + δ_{t})L(A, σ),

where ∂ denotes the map ∂ ⊗1 and δ_{t} is 1⊗ \frac{d}{dt}. For each a ∈ A\overline{m} and m ∈ Γ, let a_{m} be the image of the element a ⊗ t^{m} ∈ L(A, σ) in Alg (A, σ). These elements span Alg (A, σ), and there is a well-defined product on this space, given by

a_{m}b_{n} = ∑_{j∈ℤ_{+}} \binom{m}{j} (a_{(j)}b)_{m+n−j} ,  (0.1)

for all a ∈ A\overline{m} and b ∈ A\overline{n}.

The name twisted formal distribution algebra comes from the fact that the superalgebra Alg(A, σ) is spanned by the coefficients of the family of twisted pairwise local formal distributions

F = ⋃_{\overline{m}∈Γ/ℤ}\{a(z) = ∑_{k∈\overline{m}} a_{k}z^{−k−1} | a ∈ A\overline{m}\} .

For σ = 1 and Γ = ℤ, we recover the maximal non-twisted formal distribution superalgebra associated with the conformal superalgebra A. See [9, 10] for details.

For example, let A be an ordinary superalgebra over ℂ. The current conformal superalgebra A = ℂ[∂] ⊗_{ℂ} A is defined by letting a_{(n)}b = δ_{n,0}ab for a, b ∈ A and extending these n-products to A using the conformal superalgebra axioms. The associated loop algebra A⊗_{ℂ}ℂ[t, t^{−1}] is then encoded by the current superconformal algebra A. Taking σ to be an automorphism of A extended from a finite order (or, more generally, semisimple) automorphism of A, we recover the construction of a σ-twisted loop algebra associated to the pair (A, σ). When A is a Lie algebra, this is precisely the construction of σ-twisted loop algebras described in [8].

Under the correspondence described above, the superconformal algebras on Schwimmer and Seiberg’s lists are the Γ-twisted formal distribution algebras associated with the N = 2 and N = 4 Lie conformal superalgebras [9, 11]. Prior to Schwimmer and Seiberg’s work, it was generally assumed that the N = 2 family of superconformal algebras consisted of infinitely many distinct isomorphism classes. However, it was later recognized that this family contains (at most) two distinct isomorphism classes. A similar construction with N = 4 superconformal algebras was believed to yield an infinite family of distinct isomorphism classes [17].

The connection of the construction of the superalgebra Alg (A, σ) to the theory of differential conformal superalgebras is as follows. The ℂ[∂]-module L(A, σ) carries the structure of a differential conformal superalgebra with derivation δ = δ_{t} and n-products given by

(a ⊗ t^{k})_{(n)}(b ⊗ t^{ℓ}) = ∑_{j∈ℤ_{+}} \binom{k}{j}(a_{(n+j)}b) ⊗ t^{k+ℓ−j}.  (0.2)

Then (∂ + δ)L(A, σ) is an ideal of L(A, σ) with respect to the 0-product, which induces the product given by (0.1) on Alg (A, σ). Moreover, the differential conformal superalgebra L(A, σ) is a twisted form of the affinization L(A) = L(A, id) of A.

Thus, there are two steps to the classification of twisted superconformal algebras Alg (A, σ) up to isomorphism. First, we classify the twisted forms of the differential conformal superalgebra L(A). In light of the above discussion, this gives a complete (but possibly redundant) list of superconformal algebras, obtained by factoring by the image of ∂ + δ and retaining only the 0-product. Second, we should figure out which of these resulting superconformal algebras are non-isomorphic.

The second step of the classification is rather straightforward. For example, the twisted N = 4 superconformal algebras are distinguished by the eigenvalues of the Virasoro operator L_{0} on the odd part. The remainder of the paper will consider the first step, namely the classification of the L(A, σ) up to isomorphism.

Recently, the classification (up to isomorphism) of affine Kac-Moody algebras has been given in terms of torsors and non-abelian étale cohomology [16]. The present paper develops conformal analogues of these techniques, and lays the foundation for a classification of forms of conformal superalgebras by cohomological methods. These general results are then applied to classify the twisted N = 2 and N = 4 conformal superalgebras up to isomorphism.

To illustrate our methods, let us look at the case of the twisted loop algebras as they appear in the theory of affine Kac-Moody Lie algebras. Any such L is naturally a Lie algebra over R := ℂ[t^{±1}] and

L ⊗_{R} S ≃ 𝔤 ⊗_{ℂ} S ≃ (𝔤 ⊗_{ℂ} R) ⊗_{R} S  (0.3)

for some (unique) finite-dimensional simple Lie algebra 𝔤, and some (finite, in this case) étale extension S/R. In particular, L is an S/R-form of the R-algebra 𝔤 ⊗_{ℂ} R, with respect to the étale topology of Spec(R). Thus L corresponds to a torsor over Spec(R) under Aut(𝔤) whose isomorphism class is an element of the pointed set H_{ét}^{1}(R, Aut(𝔤)).

Similar considerations apply to forms of the R-algebra A ⊗_{k} R for any finite-dimensional algebra A over an algebraically closed field k of characteristic 0. The crucial point in the classification of forms of A ⊗_{k} R by cohomological methods is that in the exact sequence of pointed sets

H_{ét}^{1}(R, Aut^{0}(A)) → H_{ét}^{1}(R, Aut(A)) ⟶^{ψ} H_{ét}^{1}(R, Out(A)),  (0.4)

where Out(A) is the (finite constant) group of connected components of A, the map ψ is injective [16].

Grothendieck’s theory of the algebraic fundamental group allows us to identify H_{ét}^{1}(R, Out(A)) with the set of conjugacy classes of the corresponding finite (abstract) group Out(A). The injectivity of the map

H_{ét}^{1}(R, Aut(A)) ⟶^{ψ} H_{ét}^{1}(R, Out(A))

means that to any form L of A ⊗_{k} R, we can attach a conjugacy class of the finite group Out(A) that characterizes L up to R-isomorphism. In particular, if Aut(A) is connected, then all forms (and consequently, all twisted loop algebras) of A are trivial–that is, isomorphic to A ⊗_{k} R as R-algebras.

With the previous discussion as motivation, we now consider the N = 2, 4 Lie conformal superalgebras A described in [9]. The automorphism groups of these objects are as follows:

Table 1

\begin{matrix}N & Aut(A) \\ 2 & ℂ^{×} ⋊ ℤ/2ℤ \\ 4 & (SL_{2}(ℂ) × SL_{2}(ℂ))/ ± I \end{matrix}

It was originally believed that the standard N = 2 algebra lead to an infinite family of non-isomorphic superconformal algebras (arising as Γ-twisted formal distribution algebras of the different L(A, σ), as we explained above). This is somewhat surprising, for since ℂ^{×} ⋊ℤ/2ℤ has two connected components, one would expect (by analogy with the finite-dimensional case) that there would be only two non-isomorphic twisted loop algebras attached to A. Indeed, Schwimmer and Seiberg later observed that all of the superconformal algebras in one of these supposedly infinite families are isomorphic [17], and that (at most) two such isomorphism classes existed.

On the other hand, since the automorphism group of the N = 4 conformal superalgebra is connected, one would expect all twisted loop algebras in this case to be trivial and, a fortiori, that all resulting superconformal algebras would be isomorphic. Yet Schwimmer and Seiberg aver in this case the existence of an infinite family of non-isomorphic superconformal algebras!

The explanation of how, in the case of conformal superalgebras, a connected automorphism group allows for an infinite number of non-isomorphic loop algebras is perhaps the most striking consequence of our work. Briefly speaking, the crucial point is as follows. A twisted loop algebra L of a kalgebra A is always split by an extension S_{m} := k[t^{±1/m}] of R := k[t^{±1}], for some positive integer m. The extension S_{m}/R is Galois, and its Galois group can be identified with ℤ/mℤ by fixing a primitive mth root of 1 in k. The cohomology class corresponding to L can be computed using the usual Galois cohomology H^{1}(Gal(S_{m}/R), Aut(A)(S_{m})), where Aut(A)(S_{m}) is the automorphism group of the S_{m}-algebra A ⊗_{k} S_{m}. One can deal with all loop algebras at once by considering the direct limit Ŝ of the S_{m}, which plays the role of the “separable closure” of R. In fact, Ŝ is the simply connected cover of R (in the algebraic sense), and the algebraic fundamental group π_{1}(R) of R at its generic point can thus be identified with ℤ̂, namely the inverse limit of the groups Gal(S_{m}/R) = ℤ/mℤ.

Finding the “correct” definitions of conformal superalgebras over rings and of their automorphisms leads to an explanation of how Schwimmer and Seiberg’s infinite series in the N = 4 case is possible. In our framework, rings are replaced by rings equipped with a k-linear derivation (differential k-rings). The resulting concept of differential conformal superalgebra is central to our work, and one is forced to rewrite all the faithfully flat descent formalism in this setting. Under some natural finiteness conditions, we recover the situation that one encounters in the classical theory, namely that the isomorphism classes of twisted loop algebras of A are parametrized by H^{1}(ℤ̂, Aut(A)(Ŝ)), with ℤ̂ = Gal(Ŝ/R) acting continuously via automorphisms of A ⊗_{k} Ŝ.

In the N = 2 case, the automorphism group Aut(A)(Ŝ) = Ŝ^{×} ⋊ ℤ/2ℤ, and the cohomology set H^{1}(ℤ̂, Aut(A)(Ŝ)) ≃ ℤ/2ℤ, as expected. By contrast, in the N = 4 case, Aut(A)(Ŝ) is not (SL_{2}(Ŝ) × SL_{2}(Ŝ))/ ± I as we would expect from Table 1 above. In fact,

Aut(A)(Ŝ) = (SL_{2}(Ŝ) × SL_{2}(ℂ))/ ± I.

The relevant H^{1} vanishes for SL_{2}(Ŝ), but it is the somehow surprising appearance of the “constant” infinite group SL_{2}(ℂ) in which the action of π_{1}(R) = ℤ̂ is trivial that is ultimately responsible for an infinite family of non-isomorphic twisted conformal superalgebras that are parametrized by the conjugacy classes of elements of finite order of PGL_{2}(ℂ).

In this paper, we introduce differential conformal (super)algebras, and show how these can be used for the study of forms of conformal (super)algebras. However, the theory of differential conformal (super)algebras reaches far beyond. For example, it is an adequate tool in the study of differential (super)algebras; see Remark 2.7d in [9]. The ordinary conformal (super)algebras do not quite serve this purpose since they allow only translationally invariant differential (super)algebras. Another area of applicability of differential conformal (super)algebras is the theory of not necessarily translation invariant evolution PDEs.

Notation: Throughout this paper, k will be a field of characteristic zero. Unless mentioned to the contrary, we denote ⊗_{k} simply by ⊗. We will denote by k − alg the category of unital commutative associative k-algebras. If k is algebraically closed, we also fix a primitive mth root of unity ξ_{m} ∈ k for each m > 0. We assume that these roots of unity are chosen in a compatible fashion, that is, ξ_{ℓm}^{ℓ} = ξ_{m} for all positive integers ℓ and m.

The integers, nonnegative integers, and rationals will be denoted ℤ, ℤ_{+}, and ℚ, respectively. For pairs a, b of elements in a superalgebra, we let p(a, b) = (−1)^{p(a)p(b)} where p(a) (resp., p(b)) is the parity of a (resp., b).

Finally, for any linear transformation T of a given k-space V, and for any nonnegative integer n, we follow the usual convention for divided powers and define T^{(n)} := \frac{1}{n!}T^{n}.

Acknowledgments. Much of this work was completed while M.L. was an NSERC postdoc at University of Ottawa and a visiting fellow at University of Alberta. He thanks both universities for their hospitality. M.L. also thanks J. Fuchs, T. Quella, and Z. Škoda for helpful conversations. The authors are grateful to the referee for many important suggestions.

1 Conformal superalgebras

This section contains basic definitions and results about conformal superalgebras over differential rings. We recall that k denotes a field of characteristic 0, and k − alg the category of unital commutative associative k-algebras.

1.1 Differential rings

For capturing the right concept of conformal superalgebras over rings, each object in the category of base rings should come equipped with a derivation.

This leads us to consider the category k − δalg whose objects are pairs R = (R, δ_{R}) consisting of an object R of k − alg together with a k-linear derivation δ_{R} of R (a differential k–ring). A morphism from R = (R, δ_{R}) to S = (S, δ_{S}) is a k-algebra homomorphism τ : R → S that commutes with the respective derivations. That is, the diagram

\begin{matrix}R & ⟶^{τ} & S \\ ^{δ_{R}}↓ & & ↓^{δ_{S}} \\ R & ⟶^{τ} & S. \end{matrix}  (1.1)

commutes.

For a fixed R = (R, δ_{R}) as above, the objects in the category R − ext of extensions of R are the pairs (S, τ), where S = (S, δ_{S}) ∈ k − δalg and τ : R → S satisfies (1.1). Each extension (S, δ_{S}) admits an obvious Ralgebra structure: s·r = r·s := τ(r)s for all r ∈ R and s ∈ S. The morphisms in R − ext are the R-algebra homorphisms commuting with derivations. That is, for any S_{1}, S_{2} ∈ R − ext, Hom_{R−ext}(S_{1}, S_{2}) is the set of R-algebra homomorphisms in Hom_{k−δalg}(S_{1}, S_{2}).

Let S_{i} = {(S_{i}, δ_{i}) | 1 ≤ i ≤ n} be a family of extensions of R = (R, δ_{R}). Then

δ := n ; ∑_{i=1} id ⊗ ⋯ ⊗ δ_{i} ⊗ ⋯ ⊗ id

is a k-linear derivation of S_{1} ⊗_{R} S_{2} ⊗_{R} ⋯ ⊗_{R} S_{n}. The resulting extension (S_{1} ⊗_{R} S_{2} ⊗_{R} ⋯ ⊗_{R} S_{n}, δ) of (R, δ_{R}) is called the tensor product of the S_{i} and is denoted by S_{1} ⊗_{R} ⋯ ⊗_{R} S_{n}.

Similarly, we define the direct product S_{1} × ⋯ × S_{n} by considering the k-derivation δ_{1} × ⋯ × δ_{n} of S_{1} × ⋯ × S_{n}.

Example 1.2 Consider the Laurent polynomial ring R = k[t, t^{−1}]. For each positive integer m, we set S_{m} = k[t^{1/m}, t^{−1/m}] and Ŝ = \underrightarrow{lim}S_{m}.^{3} We can think of Ŝ as the ring k[t^{q} | q ∈ ℚ] spanned by all rational powers of the variable t. The k-linear derivation δ_{t} = \frac{d}{dt} of R is also a derivation of S_{m} and Ŝ. Thus R = (R, δ_{t}), S_{m} = (S_{m}, δ_{t}), and Ŝ = (Ŝ, δ_{t}) are objects in k − δalg. Clearly S_{m} is an extension of R, and Ŝ is an extension of S_{m} (hence also of R). These differential k-rings will play a crucial role in our work.

3In [3], [4], and [5], where the multivariable case is considered, the rings R, S_{m} and Ŝ were denoted by R_{1}, R_{1,m} and R_{1,∞} respectively.

1.2 Differential conformal superalgebras

Throughout this section, R = (R, δ_{R}) will denote an object of k − δalg.

Definition 1.3 An R-conformal superalgebra is a triple (A, ∂_{A}, ( −_{(n)} −)_{n∈ℤ_{+}}) consisting of

(i) a ℤ/2ℤ-graded R-module A = A\overline{0} ⊕ A\overline{1},

(ii) an element ∂_{A} ∈ End_{k}(A) stabilizing the even and odd parts of A, (iii) a k-bilinear product (a, b) ↦ a_{(n)}b for each n ∈ ℤ_{+},

satisfying the following axioms for all r ∈ R, a, b, c ∈ A and m, n ∈ ℤ_{+}:

(CS0) a_{(n)}b = 0 for n ≫ 0

(CS1) ∂_{A}(a)_{(n)}b = −na_{(n−1)}b and a_{(n)}(∂_{A}(b)) = ∂_{A}(a_{(n)}b) + na_{(n−1)}b

(CS2) ∂_{A}(ra) = r∂_{A}(a) + δ_{R}(r)a

(CS3) a_{(n)}(rb) = r(a_{(n)}b) and (ra)_{(n)}b = ∑_{j∈ℤ_{+}} δ_{R}^{(j)}(r)(a_{(n+j)}b).

If n = 0, (CS1) should be interpreted as ∂_{A}(a)_{(0)}b = 0. Note that ∂_{A} is a derivation of all n–products, called the derivation of A. The binary operation (a, b) ↦ a_{(n)}b is called the n-product of A.

If the R-conformal superalgebra A also satisfies the following two axioms, A is said to be an R-Lie conformal superalgebra:

(CS4) a_{(n)}b = −p(a, b) ∑_{j=0}^{∞}(−1)^{j+n}∂_{A}^{(j)}(b_{(n+j)}a)

(CS5) a_{(m)}(b_{(n)}c) = ∑_{j=0}^{m} \binom{m}{j}(a_{(j)}b)_{(m+n−j)}c + p(a, b)b_{(n)}(a_{(m)}c).

Remark 1.4 For a given r ∈ R we will denote the corresponding homothety a ↦ ra by r_{A}. Axiom (CS2) can then be rewritten as follows:

(CS2) ∂_{A} ∘ r_{A} = r_{A} ∘ ∂_{A} + δ_{R}(r)_{A}

Remark 1.5 If R = k, then δ_{R} is necessarily the zero derivation. Axioms (CS2) and (CS3) are then superfluous, as they simply say that ∂_{A} and the (n)-products are k-linear. The above definition thus specializes to the usual definition of conformal superalgebra over fields (cf. [9] in the case of complex numbers). Henceforth when referring to a conformal superalgebra over k, it will always be understood that k comes equipped with the trivial derivation.

Example 1.6 (Affinization of a conformal superalgebra) Let A be a conformal superalgebra over k. As in Kac [9],^{4} we define the affinization L(A) of A as follows. The underlying space of L(A) is A ⊗_{k} k[t, t^{−1}], with the ℤ/2ℤ-grading given by assigning even parity to the indeterminate t . The derivation of L(A) is

∂_{L(A)} = ∂_{A} ⊗ 1 + 1 ⊗ δ_{t}

where δ_{t} = \frac{d}{dt}, and the n-product is given by

(a ⊗ f)_{(n)}(b ⊗ g) = ∑_{j∈ℤ_{+}}(a_{(n+j)}b) ⊗ δ_{t}^{(j)}(f)g  (1.7)

for all n ∈ ℤ_{+}, a, b ∈ A, and f, g ∈ k[t, t^{−1}]. It is immediate to verify that A is a k-conformal superalgebra. In fact, A is in the obvious way a (k[t, t^{−1}], δ_{t})–conformal superalgebra.

If R = k[t, t^{−1}] and R = (R, δ_{t}) are as in Example 1.2, then the affinization L(A) = A ⊗_{k} R also admits an R-conformal structure via the natural action of R given by r′(a⊗r) := a⊗r′r for all a ∈ A and r, r′ ∈ R. The only point that needs verification is Axiom (CS2), and this is straightforward to check.

Thus A ⊗_{k} R is both a k- and an R-conformal superalgebra. We will need both of these structures in what follows. From a physics point of view, it is the k-conformal structure that matters; from a cohomological point of view, the R-conformal structure is crucial.

We will also refer to the affinization L(A) = A ⊗_{k} R as the (untwisted) loop algebra of A. It will always be made explicit whether L(A) is being viewed as a k- or as an R-conformal superalgebra.

Let A and B be R = (R, δ_{R})-conformal superalgebras. A map ϕ : A → B is called a homomorphism of R-conformal superalgebras if it is an R-module homomorphism that is homogeneous of degree \overline{0}, respects the n-products, and commutes with the action of the respective derivations. That is, it satisfies the following three properties:

(H0) ϕ is R-linear and ϕ(A\overline{ι}) ⊆ B\overline{ι} for \overline{ι} = \overline{0}, \overline{1}

(H1) ϕ(a_{(n)}b) = ϕ(a)_{(n)}ϕ(b) for all a, b ∈ A and n ∈ ℤ_{+}

(H2) ∂_{B} ∘ ϕ = ϕ ∘ ∂_{A}.

4The definition given in [9] is an adaptation of affinization of vertex algebras, as defined by Borcherds [2].

By means of these morphisms we define the category of R-conformal superalgebras, which we denote by R − conf.

An R-conformal superalgebra homomorphism ϕ : A → B is an isomorphism if it is bijective; it is an automorphism if also A = B. The set of all automorphisms of an R-conformal superalgebra A will be denoted Aut_{R−conf}(A), or simply Aut_{R}(A).

Remark 1.8 To simplify some of the longer computations, it will be convenient to use the λ-product. This is the generating function a_{λ}b defined as

a_{λ}b := ∑_{n∈ℤ_{+}} λ^{(n)}a_{(n)}b

for any pair of conformal superalgebra elements a, b and indeterminate λ, with λ^{(n)} := \frac{1}{n!}λ^{n}. In the case of Lie conformal superalgebras, we denote a_{λ}b by [a_{λ}b].

The condition (H1) that homomorphisms ϕ respect all n-products is equivalent to

(H1’) ϕ(a_{λ}b) = ϕ(a)_{λ}ϕ(b)

for all a, b.

Axiom (CS0) is equivalent to the property that a_{λ}b is polynomial in λ. Axiom (CS1) is equivalent to:

(∂_{A}(a))_{λ}b = −λa_{λ}b and a_{λ}∂_{A}b = (∂_{A} + λ)(a_{λ}b).  (1.9)

Axiom (CS2) is equivalent to

a_{λ}rb = r(a_{λ}b) and (ra)_{λ}b = (a_{λ+δ_{R}}b)_{→}r,  (1.10)

where → means that δ_{R} is moved to the right and applied to r.

Remark 1.11 Note that the homotheties r_{A} : a ↦ ra (for r ∈ R) are typically not R-conformal superalgebra homomorphisms, and that the map ∂_{A} : a ↦ ∂_{A}(a) is a R-conformal superalgebra derivation of the λ-product: ∂_{A}(a_{λ}b) = (∂_{A}a)_{λ}b + a_{λ}∂_{A}b.

1.3 Base change

Let S = (S, δ_{S}) be an extension of a base ring R = (R, δ_{R}) ∈ k − δalg. Given an R-conformal superalgebra A, the S-module A ⊗_{R} S admits an S-conformal structure, which we denote by A ⊗_{R} S, that we now describe.

The derivation ∂_{A⊗_{R}S} is given by

∂_{A⊗_{R}S}(a ⊗ s) := ∂_{A}(a) ⊗ s + a ⊗ δ_{S}(s)  (1.12)

for all a ∈ A, s ∈ S. The ℤ/2Z-grading is inherited from that of A by setting

(A ⊗_{R} S)\overline{ι} := A\overline{ι} ⊗_{R} S.

The n-products are defined via

(a ⊗ r)_{(n)}(b ⊗ s) = ∑_{j∈ℤ_{+}}(a_{(n+j)}b) ⊗ δ_{S}^{(j)}(r)s  (1.13)

for all a, b ∈ A, r, s ∈ S, and n ∈ ℤ_{+}. Axioms (CS0)–(CS3) hold, as can be verified directly. (If A is also a Lie conformal superalgebra, then (CS4)– (CS5) hold, and A ⊗_{R} S is also Lie.) The S-conformal superalgebra on A ⊗_{R} S described above is said to be obtained from A by base change from R to S.

Example 1.14 The affinization Â of a k-conformal superalgebra A (Example 1.6), viewed as a conformal superalgebra over R := (k[t, t^{−1}], δ_{t}), is obtained from A by base change from k to R.

Remark 1.15 It is straightforward to verify that the tensor products used in defining change of base are associative. More precisely, assume that S = (S, δ_{S}) is an extension of both R = (R, δ_{R}) and T = (T, δ_{T}), and U = (U, δ_{U}) is also an extension of T = (T, δ_{T}). Then for any R-conformal superalgebra A, the map (a ⊗ s) ⊗ u ↦ a ⊗ (s ⊗ u) defines a U-conformal isomorphism

(A ⊗_{R} S) ⊗_{T} U ≅ A ⊗_{R} (S ⊗_{T} U) .

Here u ∈ U acts on A ⊗_{R} (S ⊗_{T} U) by multiplication, namely

u(a ⊗ (s ⊗ u′)) := a ⊗ (s ⊗ uu′)

for a ∈ A, s ∈ S and u′ ∈ U; the derivation ∂_{A⊗(S⊗U)} acts on A⊗_{R} (S ⊗_{T} U) as

∂_{A⊗_{R}(S⊗U)} = ∂_{A} ⊗ id_{S⊗U} + id_{A} ⊗ δ_{S⊗U},

where δ_{S⊗U} := δ_{S} ⊗ id_{U} + id_{S} ⊗ δ_{U}. The associativity of tensor products will be useful when working with S/R-forms (§2 and §3 below).

Extension functor: Each extension S = (S, δ_{S}) of R = (R, δ_{R}) defines an extension functor

𝔈 = 𝔈_{S/R} : R − conf ⟶ S − conf

as follows:

Given an R-conformal superalgebra A, let 𝔈(A) be the S-conformal superalgebra A ⊗_{R} S. For each R-conformal superalgebra homomorphism ψ : A → B, the unique S-linear map satisfying

𝔈(ψ) : 𝔈(A) → 𝔈(B)  (1.16)
a ⊗ s ↦ ψ(a) ⊗ s  (1.17)

is clearly a homomorphism of S-conformal superalgebras, and it is straightforward to verify that 𝔈 is a functor.

Restriction functor: Likewise, any S-conformal superalgebra B can be viewed as an R-conformal superalgebra by restriction of scalars from S to R:

If the extension S/R corresponds to a k − δalg morphism ϕ : R → S, we view B as an R-module via ϕ. Then B is naturally an R-conformal superalgebra. The only nontrivial axiom to verify is (CS2). Using the notation of Remark 1.11, we have

∂_{B} ∘ r_{B} = ∂_{B} ∘ ϕ(r)_{B}
= ϕ(r)_{B} ∘ ∂_{B} + δ_{S}(ϕ(r))_{B}
= r_{B} ∘ ∂_{B} + ϕ(δ_{R}(r))_{B}
= r_{B} ∘ ∂_{B} + δ_{R}(r)_{B}.

This leads to the restriction functor

ℜ = ℜ_{S/R} : S − conf ⟶ R − conf,

which attaches to an S-conformal superalgebra B the same B viewed as an Rconformal superalgebra; likewise, to any S-superconformal homomorphism ψ : B → C, ℜ attaches the R-conformal superalgebra morphism ψ.

1.4 The automorphism functor of a conformal superalgebra

Let A be an R = (R, δ_{R})-conformal superalgebra. We now define the automorphism group functor Aut(A). For each extension S = (S, δ_{S}) of R, consider the group

Aut(A)(S) := Aut_{S}(A ⊗_{R} S)  (1.18)

of automorphisms of the S-conformal superalgebra A ⊗_{R} S. For each morphism ψ : S_{1} → S_{2} between two extensions S_{1} = (S_{1}, δ_{1}) and S_{2} = (S_{2}, δ_{2}) of R, and each automorphism θ ∈ Aut(A)(S_{1}), let Aut(A)(ψ)(θ) be the unique S_{2}-linear map determined by

Aut(A)(ψ)(θ) : A ⊗_{R} S_{2} → A ⊗_{R} S_{2}  (1.19)
a ⊗ 1 ↦ ∑_{i} a_{i} ⊗ ψ(s_{i})  (1.20)

for a ∈ A, where θ(a ⊗ 1) = ∑_{i} a_{i} ⊗ s_{i}.

Proposition 1.21 Aut(A) is a functor from the category of extensions of (R, δ_{R}) to the category of groups.

Proof Let θ_{1}, θ_{2} ∈ Aut_{S_{1}}(A ⊗_{R} S_{1}), and write θ_{2}(a ⊗ 1) = ∑_{i} a_{i} ⊗ s_{i} for some a_{i} ∈ A and s_{i} ∈ S_{1}. Then for any morphism ψ : S_{1} → S_{2}, we have (in the notation above): Aut(A)(ψ)(θ_{1}) ∘ Aut(A)(ψ)(θ_{2})(a ⊗ 1)

= Aut(A)(ψ)(θ_{1})(1 ⊗ ψ)θ_{2}(a ⊗ 1)
= Aut(A)(ψ)(θ_{1}) ∑_{i} a_{i} ⊗ ψ(s_{i})
= ∑_{i} ψ(s_{i})Aut(A)(ψ)(θ_{1})(a_{i} ⊗ 1)
= (1 ⊗ ψ) ∑_{i} s_{i}θ_{1}(a_{i} ⊗ 1)
= (1 ⊗ ψ)θ_{1} ∑_{i} a_{i} ⊗ s_{i}
= (1 ⊗ ψ)θ_{1}θ_{2}(a ⊗ 1)
= Aut(A)(ψ)(θ_{1}θ_{2})(a ⊗ 1).

Using the S_{2}-linearity of the S_{2}-conformal automorphisms Aut(A)(ψ)(θ_{1}), Aut(A)(ψ)(θ_{2}), and Aut(A)(ψ)(θ_{1}θ_{2}), we have

Aut(A)(ψ)(θ_{1}) ∘ Aut(A)(ψ)(θ_{2}) = Aut(A)(ψ)(θ_{1}θ_{2}).

In particular, note that for any θ ∈ Aut(A)(S_{1}), we have

Aut(A)(ψ)(θ^{−1}) ∘ Aut(A)(ψ)(θ) = Aut(A)(ψ)(id_{A⊗_{R}S_{1}}).  (1.22)

It is clear from the definition of Aut(A)(ψ) that Aut A(ψ)(id_{A⊗_{R}S_{1}}) is the identity map on A ⊗ 1, hence also on A ⊗_{R} S_{2} by S_{2}-linearity. Thus Aut(A)(ψ)(θ) is invertible, so it is bijective.

That Aut(A)(ψ)(θ) is an S_{2}-conformal superalgebra homomorphism follows easily from the assumption that θ ∈ Aut_{S_{1}}(A⊗_{R}S_{1}) and ψ : S_{1} → S_{2} is a morphism of R-extensions. Therefore, Aut(A)(ψ)(θ) ∈ Aut_{S_{2}}(A ⊗_{R} S_{2}), and we have now shown that Aut(A)(ψ) is a group homomorphism

Aut(A)(ψ) : Aut(A)(S_{1}) → Aut(A)(S_{2}).  (1.23)

Clearly Aut(A) sends the identity morphism id_{S} to the identity map on Aut(A)(S) for any extension S of R. To finish proving that Aut(A) is a functor, it remains only to note that if ψ_{1} : S_{1} → S_{2} and ψ_{2} : S_{2} → S_{3} are morphisms between extensions S_{i} = (S_{i}, δ_{i})_{1≤i≤3} of R, then for a ∈ A and θ(a ⊗ 1) = ∑_{i} a_{i} ⊗ s_{i}, we have

Aut(A)(ψ_{2}ψ_{1})(θ) : a ⊗ 1 ↦ ∑_{i} a_{i} ⊗ ψ_{2}ψ_{1}(s_{i}),

which defines precisely the same map (via S_{3}-linearity) as Aut(A)(ψ_{2}) ∘ Aut(A)(ψ_{1})(θ). Hence Aut(A)(ψ_{2}ψ_{1}) = Aut(A)(ψ_{2}) ∘ Aut(A)(ψ_{1}), which completes the proof of the proposition. □

2 Forms of conformal superalgebras and Čech cohomology

Given R in k − alg and a (not necessarily commutative, associative, or unital) R–algebra A, recall that a form of A (for the fppf–topology on Spec(R)) is an R-algebra F such that F ⊗_{R} S ≅ A ⊗_{R} S (as S-algebras) for some fppf (faithfully flat and finitely presented) extension S/R in k − alg. There is a correspondence between R-isomorphism classes of forms of A and the pointed set of non-abelian cohomology H_{fppf}^{1}(R, Aut(A)) defined à la Čech. Here Aut(A) := Aut(A)_{R} denotes the sheaf of groups over Spec(R) that attaches to an extension R′/R in k − alg the group Aut_{S}(A ⊗_{R} R′) of automorphisms of the R′–algebra A ⊗_{R} R′. For any extension S/R in k − alg, there is a canonical map

H_{fppf}^{1}(R, Aut(A)) → H_{fppf}^{1}(S, Aut(A)_{S})

The kernel of this map is denoted by H_{fppf}^{1}(S/R, Aut(A)); these are the forms of A that are trivialized by the base change S/R. One has

H_{fppf}^{1}(R, Aut(A)) = \underrightarrow{lim}H_{fppf}^{1}(S/R, Aut(A)),

where the limit is taken over all “isomorphisms classes” of fppf extensions S/R in k − alg.

In trying to recreate this construction for an R-conformal superalgebra A, we encounter a fundamental obstacle: Unlike in the case of algebras, the n-products (1.13) in A ⊗_{R} S are not obtained by S-linear extension of the n-products in A (unless the derivation of S is trivial). This prevents the automorphism functor Aut(A) from being representable in the naïve way, and the classical theory of forms cannot be applied blindly. Nonetheless, we will show in the next section that the expected correspondence between forms and cohomology continues to hold, even in the case of conformal superalgebras.

In the case of algebras, when the extension S/R is Galois, isomorphism classes of S/R–forms have an interpretation in terms of non-abelian Galois cohomology. (See [20], for instance.) We will show in §2.2 that, just as in the case of algebras, the Galois cohomology H^{1} (Gal(S/R), Aut_{S}(A ⊗_{R} S)) still parametrizes the S/R-forms of A (with the appropriate definition of Galois extension and Aut_{S}(A ⊗_{R} S)).

Throughout this section, A will denote a conformal superalgebra over R = (R, δ_{R}).

2.1 Forms split by an extension

Definition 2.1 Let S be an extension of R. An R-conformal superalgebra F is an S/R-form of A (or form of A split by S) if

F ⊗_{R} S ≅ A ⊗_{R} S

as S-conformal superalgebras.

For us, the most interesting examples of forms split by a given extension are the conformal superalgebras that are obtained via the type of twisted loop construction that one encounters in the theory of affine Kac-Moody Lie algebras.

Example 2.2 Assume k is algebraically closed. Suppose that A is a kconformal superalgebra, equipped with an automorphism σ of period m. For each i ∈ ℤ consider the eigenspace

A_{i} = {x ∈ A | σ(x) = ξ_{m}^{i}x}.

with respect to our fixed choice (ξ_{m}) of compatible primitive roots of unity in k. (The space A_{i} depends only on the class of i modulo m, of course.) Let R = k[t, t^{−1}], S_{m} = k[t^{1/m}, t^{−1/m}] and Ŝ := lim_{⟶} S_{m}. We consider the

extensions S_{m} = (S_{m}, δ_{t}) and Ŝ = (Ŝ, δ_{t}) of R = (R, δ_{t}) where δ_{t} = \frac{d}{dt}. Consider the subspace L(A, σ) ⊆ A ⊗_{k} S_{m} ⊂ A ⊗_{k} Ŝ given by

L(A, σ) = ⨁_{i∈ℤ}A_{i} ⊗ t^{i/m}.  (2.3)

Each eigenspace A_{i} is stable under ∂_{A} because σ is a conformal automorphism. From this, it easily follows that L(A, σ) is stable under the action of ∂_{A⊗S_{m}} = ∂_{A} ⊗1+1⊗δ_{t}. Since L(A, σ) is also closed under the n-products of A⊗_{k} S_{m} , it is a k-conformal subalgebra of A⊗_{k} S_{m} called the (twisted) loop algebra of A with respect to σ. Note that the definition of L(A, σ) as a subalgebra of A⊗_{k} Ŝ does not depend on the choice of the period m of the given automorphism σ. Because of this, it will be at times convenient when comparing loop algebras corresponding to automorphisms of different periods to think of loop algebras as k–conformal subalgebras of A ⊗_{k} Ŝ. It is clear that L(A, σ) is stable under the natural action of R on A ⊗_{k} S_{m}. As in Example 1.6, one checks that ∂_{A⊗S_{m}} ∘ r_{A⊗_{k}S_{m}} − r_{A⊗_{k}S_{m}} ∘ ∂_{A⊗S_{m}} = δ_{t}(r)_{A⊗_{k}S_{m}} for all r ∈ R. This shows that just as in the untwisted case, the twisted loop algebra L(A, σ) is both a k- and an R-conformal superalgebra.

Proposition 2.4 Let σ be an automorphism of period m of a k-conformal superalgebra A. Then the twisted loop algebra L(A, σ) is an S_{m}/R-form of A ⊗_{k} R.

Proof For ease of notation, we will write S = (S, δ_{S}) for S_{m} = (S_{m}, δ_{S_{m}}) in this proof. By the associativity of the tensor products used in scalar extension (Remark 1.15), the multiplication map

ψ : (A ⊗_{k} R) ⊗_{R} S → A ⊗_{k} S  (2.5)
(a ⊗ r) ⊗ s ↦ a ⊗ rs  (2.6)

(for a ∈ A, r ∈ R, and s ∈ S) is an isomorphism of S-conformal superalgebras.

Likewise, it is straightforward to verify that the multiplication map

μ : (A ⊗_{k} S) ⊗_{R} S → A ⊗_{k} S  (2.7)
(a ⊗ s_{1}) ⊗ s_{2} ↦ a ⊗ s_{1}s_{2}  (2.8)

(for a ∈ A and s_{1}, s_{2} ∈ S) is a homomorphism of S-conformal superalgebras. Indeed, μ is the composition of the “associativity isomorphism”

(A ⊗_{k} S) ⊗_{R} S → A ⊗_{k} (S ⊗_{R} S)

with the superconformal homomorphism defined by multiplication:

A ⊗_{k} (S ⊗_{R} S) → A ⊗_{k} S
a ⊗ (s_{1} ⊗ s_{2}) ↦ a ⊗ s_{1}s_{2}.

To complete the proof of Proposition 2.4, it suffices to prove that the restriction

μ : L(A, σ) ⊗_{R} S → A ⊗_{k} S  (2.9)

is bijective. For a_{i} ∈ A_{i}, we have

a_{i} ⊗ t^{j/m} = μ(a_{i} ⊗ t^{i/m} ⊗ t^{(j−i)/m}),

so μ is clearly surjective. To see that μ is also injective, assume (without loss of generality) that a k-basis {a_{λ}} of A is chosen so that each a_{λ} ∈ A_{i(λ)} for some unique 0 ≤ i(λ) < m. Let x ∈ L(A, σ) ⊗_{R} S. Since S is a free R-module with basis {t^{i/m} | 0 ≤ i ≤ m − 1}, we can uniquely write

x = ∑_{i=0}^{m−1} x_{i} ⊗ t^{i/m}

where x_{i} = ∑_{λ} a_{λ} ⊗ f_{λi} and f_{λi} ∈ t^{i(λ)/m}k[t, t^{−1}]. Then if μ(x) = 0, we have

∑_{λi} a_{λ} ⊗ f_{λi}t^{i/m} = 0,

and thus

∑_{i=0}^{m−1} f_{λ,i}t^{i/m} = 0

for all λ. Then f_{λ,i} = 0 for all λ and i. Hence x = 0, so μ is injective, and

ψ^{−1} ∘ μ : L(A, σ) ⊗_{R} S → (A ⊗_{k} R) ⊗_{R} S  (2.10)

is an S-conformal superalgebra isomorphism as desired. □

2.2 Cohomology and forms

Throughout this section S = (S, δ_{S}) will denote an extension of R = (R, δ_{R}), and A will be an R-conformal superalgebra.

Lemma 2.11 Let ψ : A → B be an R-conformal superalgebra homomorphism and γ : S → S a morphism of extensions. The canonical map

ψ ⊗ γ : A ⊗_{R} S → B ⊗_{R} S  (2.12)

is R-linear, commutes with the action of ∂_{A⊗_{R}S}, and preserves n-products. In particular, ψ ⊗ γ is an R-conformal superalgebra homomorphism via restriction of scalars from S to R:

ψ ⊗ γ : ℜ_{S/R}(A ⊗_{R} S) → ℜ_{S/R}(B ⊗_{R} S).

Proof Let x ∈ A and s ∈ S. Then

ψ ⊗ γ(∂_{A⊗_{R}S}(x ⊗ s)) = ψ ⊗ γ(∂_{A}(x) ⊗ s + x ⊗ δ_{S}(s))
= ∂_{B}(ψ(x)) ⊗ γ(s) + ψ(x) ⊗ δ_{S}(γ(s))
= ∂_{B⊗_{R}S}(ψ(x) ⊗ γ(s)).

Also, for x, y ∈ A and s, t ∈ S, we have

ψ ⊗ γ(x ⊗ s_{(n)}y ⊗ t) = ψ ⊗ γ ( ∑_{j∈ℤ_{+}} x_{(n+j)}y ⊗ δ_{S}^{(j)}(s)t)
= ∑_{j∈ℤ_{+}} ψ(x)_{(n+j)}ψ(y) ⊗ δ_{S}^{(j)}(γ(s))γ(t)
= ψ(x) ⊗ γ(s)_{(n)}ψ(y) ⊗ γ(t).
□

Corollary 2.13 The map ψ ⊗ 1 : A ⊗_{R} S → B ⊗_{R} S is an S-conformal superalgebra homomorphism.

Proof It is enough to note that the map ψ ⊗ 1 commutes with the action of S. □

For 1 ≤ i ≤ 2 and 1 ≤ j < k ≤ 3, consider the following R-linear maps:

d_{i} : S → S ⊗_{R} S
d_{jk} : S ⊗_{R} S → S ⊗_{R} S ⊗_{R} S,

defined by d_{1}(s) = s⊗1, d_{2} = 1⊗s, d_{12}(s⊗t) = s⊗t⊗1, d_{13}(s⊗t) = s⊗1⊗t, and d_{23}(s ⊗ t) = 1 ⊗ s ⊗ t for all s, t ∈ S. It is straightforward to verify that these induce R − ext morphisms d_{i} : S → S ⊗_{R} S and d_{jk} : S ⊗_{R} S → S ⊗_{R} S ⊗_{R} S. (See §1.1). Let A be an R-conformal superalgebra. By functoriality (Proposition 1.21), we obtain group homomorphisms (also denoted by d_{i} and d_{jk})

d_{i} : Aut(A)(S) → Aut(A)(S ⊗_{R} S)
d_{jk} : Aut(A)(S ⊗_{R} S) → Aut(A)(S ⊗_{R} S ⊗_{R} S)

for 1 ≤ i ≤ 2 and 1 ≤ j < k ≤ 3.

Recall that u ∈ Aut(A)(S ⊗_{R} S) is called a 1-cocycle^{5} if

d_{13}(u) = d_{23}(u)d_{12}(u).  (2.14)

On the set Z^{1}(S/R, Aut(A)) of 1-cocycles, one defines an equivalence relation by declaring two cocycles u and v to be equivalent (or cohomologous) if there exists an automorphism λ ∈ Aut(A)(S) such that

v = (d_{2}(λ)) u (d_{1}(λ))^{−1}.  (2.15)

Formula (2.15) defines an action of the group Aut(A)(S) on the set of cocycles Z^{1}(S/R, Aut(A)). The corresponding quotient set is denoted by H^{1}(S/R, Aut(A)), and this is the (nonabelian) Čech cohomology of Aut(A) relative to the covering Spec(S) → Spec(R). There is no natural group structure on the set H^{1}(S/R, Aut(A)) , but it has a distinguished element, namely the equivalence class of the identity element of the group Aut(A)(S ⊗_{R} S). We will denote this class by 1, and write [u] for the equivalence class of an arbitrary cocycle u ∈ Z^{1}(S/R, Aut(A)).

Theorem 2.16 Assume that the extension S = (S, δ_{S}) of R = (R, δ_{R}) is faithfully flat (i.e., S is a faithfully flat R-module). Then for any Rconformal superalgebra A, the pointed set H^{1}(S/R, Aut(A)) parametrizes the set of R-isomorphism classes of S/R-forms of A. Under this correspondence, the distinguished element 1 corresponds to the isomorphism class of A itself.

Proof It suffices to check that the standard descent formalism for modules is compatible with the conformal superalgebra structures.

5For faithfully flat ring extensions S/R, the 1-cocycle condition is motivated by patching data on open coverings of Spec(R). See the discussion in [20, §17.4], for instance.

Throughout this proof, fix an S/R-form B of the R-conformal superalgebra A. Let η : S⊗_{R}S → S⊗_{R}S be the “switch” map given by η(s⊗t) = t⊗s for all s, t ∈ S. Let

η_{A} := id_{A} ⊗ η : A ⊗_{R} S ⊗_{R} S → A ⊗_{R} S ⊗_{R} S
η_{B} := id_{B} ⊗ η : B ⊗_{R} S ⊗_{R} S → B ⊗_{R} S ⊗_{R} S.

We now check that the key points from the classical formalism of faithfully flat descent hold in the conformal setting:

(1) Let ψ : B ⊗_{R} S → A ⊗_{R} S be an isomorphism of S-conformal super-

algebras. Let

u_{ψ,B} := (η_{A})(ψ ⊗ 1)(η_{B})(ψ^{−1} ⊗ 1).  (2.17)

Then u_{ψ,B} ∈ Z^{1}(S/R, Aut(A)).

Proof (1): That u_{ψ,B} : A ⊗_{R} S ⊗_{R} S → A ⊗_{R} S ⊗_{R} S is S ⊗_{R} S-linear and bijective is clear, and it is straightforward to verify that u_{ψ,B} satisfies the cocycle condition (2.14). Therefore it is enough to check that u_{ψ,B} commutes with the derivation ∂_{A⊗_{R}S⊗_{R}S}, and that it preserves the n-products.

By Corollary 2.13 and the associativity of the tensor product, ψ ⊗ 1 and ψ^{−1} ⊗ 1 are S ⊗_{R} S-superconformal homomorphisms, so it is only necessary to check that η_{A} and η_{B} commute with ∂_{A⊗_{R}S⊗_{R}S} and preserve n-products. By Lemma 2.11, applied to η_{A} = id_{A} ⊗ η, it is enough to check that η commutes with δ_{S⊗S}, which is clear since

η(δ_{S⊗S}(s ⊗ t)) = η(δ_{S}(s) ⊗ t + s ⊗ δ_{S}(t))  (2.18)
= t ⊗ δ_{S}(s) + δ_{S}(t) ⊗ s  (2.19)
= δ_{S⊗S}(η(s ⊗ t))  (2.20)

for all s, t ∈ S.

(2) The class of u_{ψ,B} in H^{1}(S/R, Aut(A)) is independent of the choice of automorphism ψ in Part (1). If we denote this class by [u_{B}], then α : B ↦ [u_{B}] is a map from the set of R-isomorphism classes of S/R-forms of A to the pointed set H^{1}(S/R, Aut(A)).

Proof (2): Suppose that ϕ is another S-superconformal isomorphism

ϕ : B ⊗_{R} S → A ⊗_{R} S.

Let λ = ϕψ^{−1} ∈ Aut(A)(S). Note that d_{2}(λ)η_{A} = η_{A}d_{1}(λ). Thus

d_{2}(λ)u_{ψ,B}d_{1}(λ)^{−1} = d_{2}(λ)η_{A}(ψ ⊗ 1)η_{B}(ψ^{−1} ⊗ 1)(ψϕ^{−1} ⊗ 1)
= η_{A}(ϕψ^{−1} ⊗ 1)(ψ ⊗ 1)η_{B}(ϕ^{−1} ⊗ 1)
= u_{ϕ,B}.

(3) Let u ∈ Aut(A)(S ⊗_{R} S). Then

(a) The subset A_{u} := {x ∈ A ⊗_{R} S | u(x ⊗ 1) = η_{A}(x ⊗ 1)} is an R-conformal subalgebra of A ⊗_{R} S.

(b) The canonical map

μ_{u} : A_{u} ⊗_{R} S → A ⊗_{R} S
x ⊗ s ↦ s.x

is an S-conformal superalgebra isomorphism.

(c) If u and v are cohomologous cocycles in Z^{1}(S/R, Aut(A)), then A_{u} and A_{v} are isomorphic as R-conformal superalgebras.

Proof (3a): Clearly A_{u} is an R-submodule of A ⊗_{R} S. Next we verify that A_{u} is stable under the action of ∂_{A⊗_{R}S}.

Recall (2.18) that η_{A} commutes with the derivation ∂_{A⊗_{R}S⊗_{R}S}. Thus for all x ∈ A_{u},

u(∂_{A⊗_{R}S}(x) ⊗ 1) = u ∂_{A⊗_{R}S⊗_{R}S}(x ⊗ 1)
= ∂_{A⊗_{R}S⊗_{R}S} u(x ⊗ 1)
= ∂_{A⊗_{R}S⊗_{R}S} η_{A}(x ⊗ 1)
= η_{A}∂_{A⊗_{R}S⊗_{R}S}(x ⊗ 1)
= η_{A}∂_{A⊗_{R}S}(x) ⊗ 1.

To complete the proof of (3a), it remains only to show that A_{u} is closed under n-products. For x and y in A_{u} we have

u((x_{(n)}y) ⊗ 1) = u(x ⊗ 1_{(n)}y ⊗ 1)
= u(x ⊗ 1)_{(n)}u(y ⊗ 1)
= η_{A}(x ⊗ 1)_{(n)}η_{A}(y ⊗ 1)
= η_{A}(x ⊗ 1_{(n)}y ⊗ 1)
= η_{A}((x_{(n)}y) ⊗ 1).

Proof (3b): The map μ_{u} : A_{u} ⊗_{R} S → A ⊗_{R} S is an S-module isomorphism by the classical descent theory for modules. (See [20, Chap 17], for instance.) We need only show that it is a homomorphism of S-conformal superalgebras.

Let μ be the multiplication map

μ : A ⊗_{R} S ⊗_{R} S → A ⊗_{R} S
a ⊗ s ⊗ t ↦ a ⊗ st.

It is straightforward to verify that

μ ∘ ∂_{A⊗_{R}S⊗S} = ∂_{A⊗_{R}S} ∘ μ, and μ preserves n−products.  (2.21)

Since μ_{u} is the restriction of μ to A_{u} ⊗_{R} S it follows from (2.21) that μ_{u} preserves n-products. It remains only to show that μ_{u} commutes with ∂_{A_{u}⊗_{R}S}. But since ∂_{A_{u}⊗_{R}S} acts on A_{u} ⊗_{R} S as the restriction of the derivation ∂_{A⊗_{R}S⊗_{R}S} of A ⊗_{R} S ⊗_{R} S to the subalgebra A_{u} ⊗_{R} S and μ_{u} = μ ι, where ι is the inclusion map ι : A_{u} ⊗_{R} S ↪ A ⊗_{R} S ⊗_{R} S, we can again appeal to (2.21). This shows that μ_{u} is an S-conformal superalgebra isomorphism.

Proof (3c) Write v = d_{2}(λ) u d_{1}(λ)^{−1} for some λ ∈ Aut(A)(S). Then since d_{2}(λ) η_{A} = η_{A} d_{1}(λ), we see that

v (λ ⊗ 1)(a_{u} ⊗ 1) = v d_{1}(λ)(a_{u} ⊗ 1)
= d_{2}(λ) u(a_{u} ⊗ 1)
= d_{2}(λ) η_{A}(a_{u} ⊗ 1)
= η_{A} d_{1}(λ)(a_{u} ⊗ 1)
= η_{A} (λ ⊗ 1)(a_{u} ⊗ 1)

for all a_{u} ∈ A_{u}. Thus,

λ(A_{u}) ⊆ A_{v}.  (2.22)

Likewise, λ^{−1}(A_{v}) ⊆ A_{u}, so applying λ^{−1} to both sides of (2.22) gives:

A_{u} ⊆ λ^{−1}(A_{v}) ⊆ A_{u},

and λ(A_{u}) = A_{v}.

Since λ is an S-automorphism of A ⊗_{R} S, it commutes with the actions of S and ∂_{A⊗_{R}S}, and it preserves n-products. Thus its restriction to A_{u} commutes with R and preserves n-products. Furthermore λ∂_{A_{u}} = ∂_{A_{v}}λ since ∂_{A_{u}} and ∂_{A_{u}} are the restrictions of ∂_{A⊗_{R}S} to A_{u} and A_{v} respectively. Thus λ is an R-conformal superalgebra isomorphism from A_{u} to A_{v}.

By (3c), there is a well-defined map β : [u] ↦ [A_{u}] from the cohomology set H^{1}(S/R, Aut(A)) to the set of R-isomorphism classes of S/R-forms of A. We have also seen that β and the map α defined in (2) are inverses of each other. That the distinguished element 1 ∈ H^{1}(S/R, Aut(A)) corresponds to the algebra A is clear from the definition of A_{u} given in (3a). □

Remark 2.23 For any isomorphism ψ as in Part (1) of the proof of Theorem 2.16, the cocycle u_{ψ,B} can be rewritten in terms of the maps d_{i} : S → S ⊗_{R} S:

u_{ψ,B} = d_{2}(ψ)d_{1}(ψ)^{−1}.  (2.24)

Proof If ψ(b ⊗ 1) = ∑_{i} a_{i} ⊗ w_{i}, then

η_{A}(ψ ⊗ 1)η_{B}(b ⊗ s ⊗ t) = η_{A}(ψ ⊗ 1)(b ⊗ t ⊗ s)
= η_{A}(ψ(b ⊗ t) ⊗ s)
= η_{A}((t.ψ(b ⊗ 1)) ⊗ s)
= η_{A} (∑_{i} a_{i} ⊗ tw_{i} ⊗ s)
= ∑_{i} a_{i} ⊗ s ⊗ tw_{i}
= d_{2}(ψ)(b ⊗ s ⊗ t)

for all s, t ∈ S. Thus

u_{ψ,B} = d_{2}(ψ)(ψ^{−1} ⊗ 1)
= d_{2}(ψ)(ψ ⊗ 1)^{−1}
= d_{2}(ψ)d_{1}(ψ)^{−1}.
□

Remark 2.25 In the notation of Part (1) of the proof of Theorem 2.16, the algebra B is isomorphic to the R-conformal superalgebra B ⊗ 1 ⊆ B ⊗_{R} S by the faithful flatness of S/R. This means that there is an isomorphic copy of each S/R-form of A inside A ⊗_{R} S, and the algebra B can be recovered (up to R-conformal isomorphism) from the cocycle u_{ψ,B}, since

ψ(B ⊗ 1) = {x ∈ A ⊗_{R} S | u_{ψ,B}(x ⊗ 1) = η_{A}(x ⊗ 1)}.  (2.26)

Remark 2.27 Let S = (S, δ_{S}) be an extension of R = (R, δ_{R}). Let Γ be a finite group of automorphisms of S (as an extension of R). We say that S is a Galois extension of R with group Γ if S is a Galois extension of R with group Γ. (See [12] for definition). The unique R-module map

ψ : S ⊗_{R} S → S × ⋯ × S (|Γ| copies of S)

satisfying

a ⊗ b ↦ (γ(a)b)_{γ∈Γ}

is easily seen to be an isomorphism of R-ext (see §1.1). This induces a group isomorphism

Aut(A)(S ⊗_{R} S) ≃ Aut(A)(S) × ⋯ × Aut(A)(S)
u ↦ (u_{γ})_{γ∈Γ}

The cocycle condition u ∈ Z^{1}(S/R, Aut(A)) translates, just as in the classical situation, into the usual cocycle condition u_{γρ} = u_{γ} ^{γ}u_{ρ} where Γ acts on Aut(A)(S) = Aut_{S}(A⊗_{R} S) by conjugation, i.e., ^{γ}θ = (1⊗γ)θ(1⊗γ^{−1}). This leads to a natural isomorphism

H^{1}(S/R, Aut(A)) ≃ H^{1}(Γ, Aut_{S}(A ⊗_{R} S))

where the right-hand side is the usual Galois cohomology.

2.3 Limits

Throughout this section, R := k[t^{±1}], S_{m} := k[t^{±1/m}], Ŝ := lim_{⟶} S_{m}, and

δ_{t} := \frac{d}{dt}.

We are interested in classifying all twisted loop algebras of a given conformal superalgebra A over k. If σ ∈ Aut_{k−conf}(A) is of period m, then L(A, σ) is trivialized by the extension S_{m}/R, where R = (R, δ_{t}) and S_{m} = (S_{m}, δ_{t}). (See Example 2.2.) To compare L(A, σ) with another L(A, σ′) where σ′ is of period m′, we may consider a common refinement S_{mm′} . An elegant way of taking care of all such refinements at once is by considering the limit Ŝ = (Ŝ, δ_{t}). For algebras over k, and under some finiteness assumptions, Ŝ plays the role of the separable closure of R (see [3] and [5] for details). We follow this philosophy in the present situation.

Let m ∈ ℤ_{+}, and let : ℤ → ℤ/mℤ be the canonical map. Each extension S_{m}/R is Galois with Galois group ℤ/mℤ, where \overline{1}(t^{1/m}) = ξ_{m}t^{1/m}. (Our choice of roots of unity ensures that the action of ℤ/ℓmℤ on S_{m} is compatible with that of ℤ/mℤ under the canonical map ℤ/ℓmℤ → ℤ/mℤ.) Fix an algebraic closure \overline{k(t)} of k(t) containing all of the rings S_{m}, and let π_{1}(R) be the algebraic fundamental group of Spec(R) at the geometric point a = Spec ( \overline{k(t)} ). (See [18] for details.) Then Ŝ is the algebraic simplyconnected cover of R (in the algebraic sense), and π_{1}(R) = ℤ̂ := lim_{⟵} ℤ/mℤ,

where ℤ̂ acts continuously on Ŝ via ^{1}t^{p/q} = ξ_{q}^{p}t^{p/q}. Let π : π_{1}(R) → Aut_{R}(Ŝ) be the corresponding group homomorphism.

Let A be an R = (R, δ_{t})-conformal superalgebra. As in Remark 2.27, π_{1}(R) acts on Aut(A)(Ŝ) = AutŜ(A ⊗_{R} Ŝ) by means of π. That is, if γ ∈ π_{1}(R) and θ ∈ AutŜ(A ⊗_{R} Ŝ), then

γθ = (1 ⊗ π(γ))θ(1 ⊗ π(γ)^{−1}).

Let A be an R-conformal superalgebra. Given another R-conformal superalgebra B, we have

\begin{aligned}A ⊗_{R} S_{m} ≃_{S_{m}} B ⊗_{R} S_{m} & ⟹ A ⊗_{R} S_{m} ⊗_{S_{m}} S ≃_{S} B ⊗_{R} S_{m} ⊗_{S_{m}} S \\ & ⟹ A ⊗_{R} S ≃_{S} B ⊗_{R} S \end{aligned}

for all extensions S of S_{m}. This yields inclusions

H^{1}(S_{m}/R, Aut(A)) ⊆ H_{1}(S_{n}/R, Aut(A)) ⊆ H^{1}(Ŝ/R, Aut(A))

for all m|n, hence a natural injective map

η : lim_{⟶} H^{1}(S_{m}/R, Aut(A)) → H^{1}(Ŝ/R, Aut(A)).  (2.28)

In the classical situation, namely when A is an algebra, the surjectivity of the map η is a delicate problem (see [14] for details and references). The following result addresses this issue for an important class of conformal superalgebras.

Proposition 2.29 Assume that A satisfies the following finiteness condition:

(Fin) There exist a_{1}, …, a_{n} ∈ A such that the set {∂_{A}^{ℓ}(ra_{i}) | r ∈ R, ℓ ≥ 0} spans A.

Then.

(1) For all R-superconformal algebra B the natural map

lim_{⟶} Hom_{S_{m}}(A ⊗_{R} S_{m}, B ⊗_{R} S_{m}) → HomŜ(A ⊗_{R} Ŝ, B ⊗_{R} Ŝ) is bijective.

(2) The natural map η : lim_{⟶} H^{1}(S_{m}/R, Aut(A)) → H^{1}(Ŝ/R, Aut(A)) is bijective.

(3) The action of the profinite group π_{1}(R) acts continuously on AutŜ(A⊗_{R} Ŝ) = Aut(A)(Ŝ) and

H^{1}(Ŝ/R, Aut(A)) ≃ H_{ct}^{1}(π_{1}(R), Aut (A)(Ŝ)),

where the right H_{ct}^{1} denotes the continuous non-abelian cohomology of the profinite group π_{1}(R) acting (continuously) on the group Aut (A)(Ŝ)

Proof (1) To establish the surjectivity of our map we must show that every Ŝ-conformal superalgebra homomorphism

ψ : A ⊗_{R} Ŝ → B ⊗_{R} Ŝ

is obtained by base change from an S_{m}-homomorphism

ψ_{m} : A ⊗_{R} S_{m} → B ⊗_{R} S_{m}.

Let m > 0 be sufficiently large so that ψ(a_{i} ⊗ 1) ∈ B ⊗_{R} S_{m} for all i. Since ∂_{B⊗_{R} Ŝ} = ∂_{B} ⊗ 1 + 1 ⊗ δ_{t} stabilizes B ⊗_{R} S_{m}, we have

ψ(∂_{A}^{ℓ}(ra_{i}) ⊗ 1) = ψ(∂_{A⊗_{R} Ŝ}^{ℓ}(ra_{i} ⊗ 1))
= ∂_{B⊗_{R} Ŝ}^{ℓ}(ψ(ra_{i} ⊗ 1)) = ∂_{B⊗_{R} Ŝ}^{ℓ}(ψ(a_{i} ⊗ r))
∈ ∂_{B⊗_{R} Ŝ}^{ℓ}(B ⊗_{R} S_{m}) ⊆ B ⊗_{R} S_{m} .

Thus ψ(A ⊗ 1) ⊆ B ⊗_{R} S_{m}. Since ψ is S_{m}-linear, we get

ψ(A ⊗_{R} S_{m}) ⊆ B ⊗_{R} S_{m} .

By restriction, we then have an S_{m}-conformal superalgebra homomorphism

ψ_{m} : A ⊗_{R} S_{m} → B ⊗_{R} S_{m},

which by base change induces ψ.

As for the injectivity, let i = 1, 2 and consider two elements ψ_{i} ∈ Hom_{S_{m_{i}}} (A⊗_{R}S_{m_{i}}, B⊗_{R}S_{m_{i}}) that map to the same element ψ of HomŜ(A⊗_{R} Ŝ, B ⊗_{R} Ŝ). Let m = m_{1}m_{2} and consider the images ψ′_{i} of the ψ_{i} in Hom_{S_{m}}(A ⊗_{R} S_{m}, B ⊗_{R} S_{m}). Then both ψ′_{1} and ψ′_{2} map to ψ via the base change S_{m} → Ŝ. Since the extension Ŝ/S_{m} in k-alg is faithfully flat we conclude that ψ′_{1} = ψ′_{2}. This shows that ψ_{1} and ψ_{2} yield the same element of

lim_{⟶} Hom_{S_{m}}(A ⊗_{R} S_{m}, B ⊗_{R} S_{m}).

(2) If we think of the relevant H^{1} as measuring isomorphism classes of forms, then that the map η is bijective is an easy application of (1).

(3) The reasoning of (1) above shows that an automorphism ψ of the Ŝ-conformal superalgebra A ⊗_{R} Ŝ is determined by its values on the a_{i} ⊗ 1. Since Ŝ is a free R-module admitting the set {t^{q} : 0 ≤ q < 1}_{q∈ℚ} as a basis we can write ψ(a_{i} ⊗ 1) = Σa_{q}(i) ⊗ t^{q} for some unique a_{q}(i) ∈ A. It follows that for γ ∈ ℤ̂ = π_{1}(R) we have ^{γ}ψ = ψ if and only if ^{γ}(t^{q}) = t^{q} whenever a_{q}(i) ≠ 0. Since the action of π_{1}(R) on Ŝ is continuous the stabilizer in π_{1}(R) of each t^{q} is open. Thus the stabilizer of ψ in π_{1}(R) is a finite intersection of open subgroups, hence open. This shows that the action of π_{1}(R) on Aut(A)(Ŝ) is continuous. Now (3) follows from (2) and Remark 2.27 □

Remark 2.30 We shall later see that all of the conformal superalgebras that interest us do satisfy the above finiteness condition. Computing H_{ct}^{1}(π_{1}(R), Aut (A)(Ŝ)) is thus central to the classification of forms. The following two results are therefore quite useful.

(1) If G is a linear algebraic group over k whose identity connected component is reductive, then the canonical map

H_{ct}^{1}(π_{1}(R), G(Ŝ)) → H_{ét}^{1}(R, G)

is bijective.

(2) If 𝔊 is a reductive group scheme over Spec(R) then H_{ét}^{1}(R, 𝔊) = 1.

The first result follows from Corollary 2.16.3 of [5], while (2) is the main result of [16].

2.4 The centroid trick

Analogous to work done for Lie (super)algebras [1, 6], it is possible to study the more delicate question of k-conformal isomorphism, as opposed to the stronger condition of R-conformal isomorphism, using a technique that we will call the centroid trick.^{6} In this section, we collect some general facts about centroids and the relationship between these two types of conformal isomorphism. In the next section, we will apply the results of this section to some interesting examples.

Except where otherwise explicitly noted, we assume throughout §2.4 that R = (R, δ_{R}) is an arbitrary object of k − δalg. Recall that if R = k, then δ_{k} = 0.

6This name was suggested by B.N. Allison to emphasize the idea’s widespread applicability.

For any R-conformal superalgebra A, let Ctd_{R}(A) be the set

{χ ∈ End_{R−smod}(A) | χ(a_{(n)}b) = a_{(n)}χ(b) for all a, b ∈ A, n ∈ ℤ_{+}},

where End_{R−smod}(A) is the set of homogeneous R-supermodule endomorphisms A → A of degree \overline{0}.

Recall that for r ∈ R we use r_{A} to denote the homothety a ↦ ra. By Axiom (CS3), we have r_{A} ∈ Ctd_{R}(A). Let R_{A} = {r_{A} : r ∈ R}. We have a canonical morphism of associative k- (and R-) algebras

R → R_{A} ⊆ Ctd_{R}(A).  (2.31)

Via restriction of scalars, our R-conformal superalgebra A admits a kconformal structure (where, again, k is viewed as an object of k − δalg by attaching the zero derivation). This yields the inclusion

Ctd_{R}(A) ⊆ Ctd_{k}(A).  (2.32)

Lemma 2.33 Let A and B be R-conformal superalgebras such that their restrictions are isomorphic k-conformal superalgebras. That is, suppose there is a k-conformal superalgebra isomorphism ϕ : A → B (where A and B are viewed as k-conformal superalgebras by restriction). Then the following properties hold.

(i) The map χ → ϕχϕ^{−1} defines an associative k-algebra isomorphism Ctd(ϕ) : Ctd_{k}(A) → Ctd_{k}(B). Moreover, ϕ is an R-conformal superalgebra isomorphism if and only if Ctd(ϕ)(r_{A}) = r_{B} for all r ∈ R.

(ii) The map δ_{Ctd_{k}(A)} : χ ↦ [∂_{A}, χ] = ∂_{A}χ − χ∂_{A} is a derivation of the associative k-algebra Ctd_{k}(A). Furthermore, the diagram

Ctd_{k}(A) ⟶^{Ctd(ϕ)} Ctd_{k}(B)
δ_{Ctd_{k}(A)}↓  ↓^{δ_{Ctd_{k}(B)}}
Ctd_{k}(A) ⟶^{Ctd(ϕ)} Ctd_{k}(B).

(2.34)

commutes.

Proof (i) This is a straightforward consequence of the various definitions.

(ii) For any χ ∈ Ctd_{k}(A), a, b ∈ A and n ≥ 0,

[∂_{A}, χ] (a_{(n)}b) = (∂_{A}χ − χ∂_{A}) (a_{(n)}b)
= ∂_{A} (a_{(n)}χ(b)) − χ (∂_{A}(a)_{(n)}b + a_{(n)}∂_{A}(b))

= ∂_{A}(a)_{(n)}χ(b) + a_{(n)}∂_{A}(χ(b)) − ∂_{A}(a)_{(n)}(χb) − a_{(n)}χ(∂_{A}(b))

= a_{(n)}[∂_{A}, χ](b).

The commutativity of Diagram 2.34 is easy to verify. □

For some of the algebras which interest us the most (e.g. the conformal superalgebras in §3), the natural ring homomorphisms R → Ctd_{k}(A) are isomorphisms. This makes the following result relevant.

Proposition 2.35 Let A_{1} and A_{2} be conformal superalgebras over R = (R, δ_{R}). Assume that Aut_{k}(R) = 1, i.e., the only k-algebra automorphism of R that commutes with the derivation δ_{R} is the identity. Also assume that the canonical maps R → Ctd_{k}(A_{i}) are k-algebra isomorphisms for i = 1, 2. Then A_{1} and A_{2} are isomorphic as k-conformal superalgebras if and only if they are isomorphic as R-conformal superalgebras.

Proof Clearly, if ϕ : A_{1} → A_{2} is an isomorphism of R-conformal superalgebras, then it is also an isomorphism of k-conformal superalgebras when A_{1} and A_{2} are viewed as k-conformal superalgebras by restriction of scalars.

Now suppose that ϕ : A_{1} → A_{2} is a k-conformal isomorphism, and consider the resulting k-algebra isomorphism Ctd(ϕ) : Ctd_{k}(A_{1}) → Ctd_{k}(A_{2}) of Lemma 2.33. Under the identification R_{A_{1}} = R = R_{A_{2}}, the commutativity of Diagram (2.34), together with [∂_{A_{1}}, r_{A_{1}}] = δ_{R}(r)_{A_{1}} and [∂_{A_{2}}, r_{A_{2}}] = δ_{R}(r)_{A_{2}}, yields that Ctd(ϕ), when viewed as an element of Aut_{k−alg}(R), commutes with the action of δ_{R}. By hypothesis, Ctd(ϕ) = id_{R}, and therefore ϕ is R-linear by Lemma 2.33(i). Since ϕ also commutes with ∂_{A_{1}} and preserves n-products, it is an R-conformal isomorphism. □

Corollary 2.36 Let A_{1} and A_{2} be conformal superalgebras over R = (R, ∂_{t}) where R = k[t, t^{−1}] and δ_{t} = \frac{d}{dt}. Assume that the canonical maps R → Ctd_{k}(A_{i}) are k-algebra isomorphisms, for i = 1, 2. Then

A_{1} ≅_{k} A_{2} if and only if A_{1} ≅_{R} A_{2}.

Proof The only associative k-algebra automorphisms of R are given by maps t ↦ αt^{ϵ}, where α is a nonzero element of k and ϵ = ±1. Thus the only k-algebra automorphism commuting with the derivation δ_{t} is the identity map, so the conditions of Proposition 2.35 are satisfied. □

Remark 2.37 The previous corollary is in sharp contrast to the situation that arises in the case of twisted loop algebras of finite-dimensional simple Lie algebras, where k-isomorphic forms need not be R-isomorphic. (See [1] and [16].) The rigidity encountered in the conformal case is due to the presence of the derivation δ_{t}.

2.5 Central extensions

In this section we assume that our conformal superalgebras are Lie-conformal, i.e. they satisfy axioms (CS4) and (CS5). Following standard practice we denote the λ-product a_{λ}b by [a_{λ}b] (which is then called the λ-bracket.)

A central extension of a k-conformal superalgebra A is a k-conformal superalgebra Ã and a conformal epimorphism π : Ã → A with kernel ker π contained in the centre Z(Ã) = {a ∈ Ã | [a_{λ}b] = 0 for all b ∈ Ã} of Ã. Given two central extensions (Ã, π) and (B̃, μ) of A, a morphism (from (Ã, π) to (B̃, μ)) in the category of central extensions is a conformal homomorphism ϕ : Ã → B̃ such that μ ∘ ϕ = π. A central extension of A is universal if there is a unique morphism from it to every other central extension of A.

Proposition 2.38 Let (Ã_{i}, π_{i}) be central extensions of k-conformal superalgebras A_{i} with ker π_{i} = Z(Ã_{i}) for i = 1, 2. Suppose ψ̃ : Ã_{1} → Ã_{2} is a k-conformal isomorphism. Then A_{1} ≅_{k−conf} A_{2}.

Proof In the category of k-vector spaces, fix sections σ_{i} : A_{i} → Ã_{i} of π_{i} : Ã_{i} → A_{i}. We verify that

ψ = π_{2} ∘ ψ̃ ∘ σ_{1} : A_{1} → A_{2}

is a k-conformal isomorphism.

Note that

π_{1} [σ_{1}(x)_{λ}σ(y)] = [π_{1}σ_{1}(x)_{λ}π_{1}σ_{1}(y)]
= [x_{λ}y]
= π_{1}(σ_{1} [x_{λ}y]),

so σ_{1} [x_{λ}y] = [σ_{1}(x)_{λ}σ_{1}(y)] + w(λ) for some polynomial w(λ) in the formal variable λ with coefficients in ker π_{1} = Z(Ã_{1}).

Moreover, [ψ̃(u)_{λ}ψ̃(a)] = ψ̃[u_{λ}a] = 0, for all u ∈ Z(Ã_{1}) and a ∈ Ã_{1}, so

it is easy to see that ψ̃ restricts to a bijection between Z(Ã_{1}) and Z(Ã_{2}).

Therefore,

ψ [x_{λ}y] = π_{2} ∘ ψ̃ ∘ σ_{1} [x_{λ}y]
= π_{2} ∘ ψ̃( [σ_{1}(x)_{λ}σ_{1}(y)] )
= π_{2} ∘ ψ̃[σ_{1}(x)_{λ}σ_{1}(y)]
= [π_{2} ∘ ψ̃ ∘ σ_{1}(x)_{λ}π_{2} ∘ ψ̃ ∘ σ_{1}(y)]
= [ψ(x)_{λ}ψ(y)] .

To see that ψ commutes with the derivations, note that π_{1}(∂Ã_{1}(x)) =

∂_{A_{1}}(π_{1}(x)) for all x ∈ A_{1}. Thus π_{1}(∂Ã_{1}(σ_{1}(x))) = ∂_{A_{1}}(π_{1}σ_{1}(x)) = ∂_{A_{1}}(x), so ∂Ã_{1}(σ_{1}(x)) = σ_{1}(∂_{A_{1}}(x)) + u for some u ∈ ker π_{1}. Hence

∂_{A_{2}}ψ(x) = ∂_{A_{2}}π_{2} ∘ ψ̃ ∘ σ_{1}(x)
= π_{2} ∘ ψ̃(∂Ã_{1}σ_{1}(x))
= π_{2} ∘ ψ̃ ∘ σ_{1}(∂_{A_{1}}(x)) + π_{2} ∘ ψ̃(u)
= ψ(∂_{A_{1}}(x))

since ψ̃ : Z(Ã_{1}) → Z(Ã_{2}). Hence ψ : A_{1} → A_{2} is a homomorphism of k-conformal superalgebras.

The map ψ is clearly injective: if x ∈ ker ψ, then ψ̃ ∘ ψ_{1}(x) ∈ ker π_{2} = Z(A_{2}), so σ_{1}(x) ∈ Z(A_{1}) and x = π_{1}ψ_{1}(x) = 0.

To see that ψ is surjective, let y ∈ A_{2}. Then let x = π_{1} ∘ ψ̃^{−1} ∘ σ_{2}(y). For any x̃ ∈ Ã_{1}, we have σ_{1}π_{1}(x̃) = x̃ + v for some v ∈ ker π_{1}. Thus

ψ(x) = π_{2} ∘ ψ̃(σ_{1} ∘ π_{1}(ψ̃^{−1} ∘ σ_{2}(y)))
= π_{2} ∘ ψ̃(ψ̃^{−1} ∘ σ_{2}(y) + v)

for some v ∈ ker π_{1}. Then

ψ(x) = π_{2} ∘ σ_{2}(y) + π_{2} ∘ ψ̃(v) = y,

and ψ : A_{1} → A_{2} is surjective. Hence ψ is an isomorphism of k-conformal superalgebras. □

Corollary 2.39 Suppose that (Ã_{i}, π_{i}) are universal central extensions of k-conformal superalgebras A_{i} with Z(A_{i}) = 0 for i = 1, 2. Then

Ã_{1} ≅_{k−conf} Ã_{2} if and only if A_{1} ≅_{k−conf} A_{2}.
□

3 Examples and applications

In this section, we compute the automorphism groups of some important conformal superalgebras. It is then easy to explicitly classify forms of these algebras by computing the relevant cohomology sets introduced in §2.2.

For all of this section, we fix the notation

R = (R, δ_{R}) := (k[t, t^{−1}], \frac{d}{dt})
Ŝ = (Ŝ, δŜ) := (k[t^{q} | q ∈ ℚ], \frac{d}{dt}) ,

where k is an algebraically closed field of characteristic zero.

By definition, every k-conformal superalgebra is a ℤ/2ℤ-graded module A over the polynomial ring k[∂] where ∂ acts on A via ∂_{A}. For the applications below, we work with k-conformal superalgebras A which are free k[∂]-supermodules. That is, there exists a ℤ/2ℤ-graded subspace V = V\overline{0} ⊕ V\overline{1} ⊆ A so that

A\overline{ι} = k[∂] ⊗_{k} V\overline{ι}

for \overline{ι} = \overline{0}, \overline{1}.

The following result is extremely useful in computing automorphism groups of conformal superalgebras.

Lemma 3.1 Let A = k[∂] ⊗_{k} V be a k-conformal superalgebra which is a free k[∂]-supermodule. Let S = (S, δ_{S}) be an arbitrary object of k − δalg. Then

(i) Every automorphism of the S-conformal superalgebra A ⊗_{k} S is completely determined by its restriction to V ≃ (1 ⊗ V) ⊗ 1 ⊆ A ⊗_{k} S.

(ii) Assume ϕ : V ⊗_{k} S → V ⊗_{k} S is a bijective parity-preserving S-linear map such that ϕ([v ⊗ 1_{λ}w ⊗ 1]) = [ϕ(v ⊗ 1)_{λ}ϕ(w ⊗ 1)] for all v, s ∈ V. Then there is a unique automorphism ϕ̂ ∈ Aut(A)(S) extending ϕ.

Proof Let {v_{i} | i ∈ I} be a k-basis of V consisting of homogeneous elements relative to its ℤ/2ℤ-grading, and {s_{j} | j ∈ J} a k-basis of S. Since ∂_{A⊗S} = ∂_{A} ⊗ 1 + 1 ⊗ δ_{S} and since the v_{i} form a k[∂]-basis of A, the set

{∂_{A⊗S}^{ℓ}(v_{i} ⊗ s_{j}) | i ∈ I, j ∈ J, ℓ ≥ 0}

is a k-basis of A ⊗_{k} S. Because any S-automorphism must commute with ∂_{A⊗S}, we have no choice but to define ϕ̂ to be the unique k-linear map on the vector space A ⊗_{k} S satisfying ϕ̂(∂_{A⊗S}^{ℓ}(v_{i} ⊗ s_{j})) = ∂_{A⊗S}^{ℓ}(ϕ(v_{i} ⊗ s_{j})). In particular, we have

ϕ̂(∂_{A⊗S}^{ℓ}(x)) = ∂_{A⊗S}^{ℓ}(ϕ(x)).  (3.2)

for all x ∈ V ⊗_{k} S. We claim that ϕ̂ ∈ Aut(A)(S). It is immediate from the definition that ϕ̂ is invertible, and that it commutes with the action of ∂_{A⊗S}.

For any v, w ∈ V, r, s ∈ R, and n ∈ ℤ_{+},

ϕ(v ⊗ r_{(n)}w ⊗ s) = sϕ(v ⊗ r_{(n)}w ⊗ 1)

= −s p(v, w) ∑_{j=0}^{∞}(−1)^{n+j}∂_{A⊗S}^{(j)}ϕ(w ⊗ 1_{(n+j)}v ⊗ r)

= −s p(v, w) ∑_{j=0}^{∞}(−1)^{n+j}∂_{A⊗S}^{(j)}rϕ(w ⊗ 1_{(n+j)}v ⊗ 1)

= −s p(v, w) ∑_{j=0}^{∞}(−1)^{n+j}∂_{A⊗S}^{(j)}rϕ(w ⊗ 1)_{(n+j)}ϕ(v ⊗ 1)

= −s p(v, w) ∑_{j=0}^{∞}(−1)^{n+j}∂_{A⊗S}^{(j)}ϕ(w ⊗ 1)_{(n+j)}rϕ(v ⊗ 1)

= s (rϕ(v ⊗ 1))_{(n)}ϕ(w ⊗ 1)

= ϕ(v ⊗ r)_{(n)}ϕ(w ⊗ s),

by (CS4) and (CS3). Similar arguments using (CS1) and (CS4) show that for any homogeneous x, y ∈ V ⊗ S and ℓ, m, n ∈ ℤ_{+},

\begin{aligned}ϕ̂(∂_{A⊗S}^{(m)}(x)_{(n)}∂_{A⊗S}^{(ℓ)}(y)) & = ∂_{A⊗S}^{(m)}ϕ(x)_{(n)}∂_{A⊗S}^{(ℓ)}ϕ(y) \\ & = ϕ̂(∂_{A⊗S}^{(m)}(x))_{(n)}ϕ̂(∂_{A⊗S}^{(ℓ)}(y)), \end{aligned}

so ϕ̂ preserves n-products. Finally, to see that ϕ̂ is S-linear, we first observe that

s_{A⊗S} ∘ ∂_{A⊗S}^{(n)} = n ; ∑_{i=0}(−1)^{i}∂_{A⊗S}^{(i)} ∘ δ_{S}(s)_{A⊗S}^{(n−i)}.  (3.3)

This follows from repeated use of Axiom (CS2) applied to the S-conformal superalgebra A ⊗_{k} S when taking into account that ∂_{A⊗S} = ∂_{A} ⊗ 1 + 1 ⊗ δ_{S}.

For all s ∈ S and x ∈ V ⊗_{k} S, we then have

ϕ̂(s_{A⊗S} ∘ ∂_{A⊗S}^{(n)}(x))
= ϕ̂( ∑_{i=0}^{n}(−1)^{i}∂_{A⊗S}^{(n)} ∘ δ_{S}(s)_{A⊗S}^{(n−i)}(x)))  (by 3.3)

= ∑_{i=0}^{n}(−1)^{i}∂_{A⊗S}^{(n)} ϕ̂(δ_{S}(s)_{A⊗S}^{(n−i)}(x)))  (by definition of ϕ̂)

= ∑_{i=0}^{n}(−1)^{i}∂_{A⊗S}^{(n)} ∘ δ_{S}(s)_{A⊗S}^{(n−i)}ϕ(x))  (S − linearity on V ⊗_{k} S)
= s_{A⊗S} ∘ ∂_{A⊗S}^{(n)}ϕ(x)  (by 3.3)
= s_{A⊗S} ∘ ϕ̂(∂_{A⊗S}^{(n)}(x)),  (by definition of ϕ̂)

so ϕ̂ commutes with the operator s_{A⊗S}, and ϕ̂ is thus S-linear. □

3.1 Current conformal superalgebras

Let V = V\overline{0} ⊕ V\overline{1} be a Lie superalgebra of arbitrary dimension over the field k. Let Curr V be the current conformal superalgebra

Curr (V) := k[∂] ⊗_{k} V

with n-products defined by the λ-bracket^{7} [v_{λ}w] = [v, w], where [v, w] is the Lie superbracket for all v, w ∈ V. The derivation ∂_{Curr (V)} is given by the natural action of ∂ on the k-space Curr (V).

Theorem 3.4 Let V be a semisimple Lie superalgebra over k, and let σ ∈ Aut_{k−conf}(Curr V). Then σ(V) ⊆ V.

Proof Let A = Curr V, and let π_{V} : A → V = k ⊗_{k} V be the projection of A onto the first component of the (vector space) direct sum

A = (k ⊗_{k} V) ⊕ (∂k[∂] ⊗_{k} V).

Let σ_{V} = π_{V} σ : V → V, and extend σ_{V} to A by k[∂]-linearity.

To verify that σ_{V} is a k-conformal homomorphism, we expand both sides of the following equation for all x, y ∈ V:

σ [x_{λ}y] = [σ(x)_{λ}σ(y)] .  (3.5)

7See Remark 1.8.

The left-hand side expands as

σ [x_{λ}y] = σ_{V} [x_{λ}y] + (σ − σ_{V}) [x_{λ}y] .  (3.6)

The right-hand side is

[σ(x)_{λ}σ(y)] = [σ_{V} (x)_{λ}σ_{V} (y)] + [σ_{V} (x)_{λ}(σ − σ_{V})(y)]
+ [(σ − σ_{V})(x)_{λ}σ_{V} (y)] + [(σ − σ_{V})(x)_{λ}(σ − σ_{V})(y)] . (3.7)

By (1.9),

[σ_{V} (x)_{λ}(σ − σ_{V})(y)] + [(σ − σ_{V})(x)_{λ}σ_{V} (y)] + [(σ − σ_{V})(x)_{λ}(σ − σ_{V})(y)]

is contained in the space

k[λ] ⊗ ∂k[∂] ⊗ V + λk[λ] ⊗ k[∂] ⊗ V,

as is (σ − σ_{V}) [x_{λ}y]. Therefore, we can apply π_{V} to the right-hand sides of (3.6) and (3.7) and then evaluate at λ = 0 to obtain

σ_{V} [x_{λ}y] = [σ_{V} (x)_{λ}σ_{V} (y)] .  (3.8)

Thus σ_{V} : A → A is a homomorphism of k-conformal superalgebras.

In fact, σ_{V} is a k-conformal superalgebra isomorphism. By the k[∂]- linearity of σ_{V}, it is sufficient to verify that its restriction σ_{V} : V → V is bijective. But this is straightforward: if σ_{V} (x) = 0, then σ(x) = ∂a for some a ∈ A, and x = ∂σ^{−1}(a). But x ∈ V, so x = 0, and σ_{V} is injective. Likewise, if y ∈ V, then write σ^{−1}(y) = z + ∂b for some z ∈ V and b ∈ A. Then

y = σ_{V} σ^{−1}(y)
= σ_{V} (z) + σ_{V} (∂b)
= σ_{V} (z) + π_{V} (∂σ(b))
= σ_{V} (z),

so σ_{V} is also surjective. Hence σ_{V} : A → A is a k-conformal automorphism.

Therefore the map

τ = σ_{V}^{−1}σ : A → A  (3.9)

is a k-conformal automorphism. Note that π_{V} τ(x) = x for all x ∈ V. Since σ = σ_{V} τ and σ_{V} (V) ⊆ V, Theorem 3.4 will be proven if we show that τ(V) ⊆ V.

For nonzero x ∈ V write

τ(x) = ∑_{i=0}^{M(x)} ∂^{(i)}v_{ix}  (3.10)

for some v_{ix} ∈ V, with v_{M(x),x} ≠ 0. Define v_{i0} to be zero for all i and M(0) = −1. Let

W = Span_{k}{v_{M(x),x} | x ∈ V }.  (3.11)

We claim that W ⊆ V is an ideal of the Lie algebra V. Indeed, for any x, y ∈ V,

τ[x, y] = τ [x_{λ}y]

= [τ(x)_{λ}τ(y)]

= [ ∑_{i=0}^{M(x)} ∂^{(i)}v_{ix} _{λ} ∑_{j=0}^{M(y)} ∂^{(j)}v_{jy}]

= ∑_{i=0}^{M(x)} ∑_{j=0}^{M(y)}(−λ)^{(i)}(∂ + λ)^{(j)}[v_{ix}, v_{jy}].  (3.12)

The highest power of ∂ in (3.12) is ∂^{(M(y))}. Since the indeterminate λ does not occur in the expression τ[x, y], we see that either

M([x, y]) = M(y)  (3.13)

or else

[v_{0x}, v_{M(y),y}] = 0.  (3.14)

If (3.13) holds, then v_{M([x,y]),[x,y]} = [v_{0x}, v_{M(y),y}], so

[x, v_{M(y),y}] ∈ W  (3.15)

since v_{0x} = π_{V} τ(x) = x. If (3.14) holds, then (3.15) holds trivially since [x, v_{M(y),y}] = 0. Therefore [V, v_{M(y),y}] ⊆ W for all y ∈ V, so W is an ideal of the Lie superalgebra V.

Suppose that M(x) > 0 for some x. Comparing the highest powers of λ occurring on both sides of (3.12), we have

0 = (−λ)^{(M(x))}λ^{(M(y))}[v_{M(x),x}, v_{M(y),y}]

for all y ∈ V. That is, [v_{M(x),x}, v_{M(y),y}] = 0 for all y ∈ V, so v_{M(x),x} is in the centre Z(W) of the Lie superalgebra W. But Z(W) is an ideal of V. (This follows easily from the Jacobi identity.) Since Z(W) is solvable and V is semisimple, we see that Z(W) = 0 and v_{M(x),x} = 0, a contradiction. Hence M(x) = 0 for all nonzero x ∈ V. Therefore, τ(x) = v_{0x} = π_{V} τ(x) = x for all x ∈ V, so the k[∂]-linear map τ is the identity map on A, and σ = σ_{V} τ = σ_{V} : V → V. □

Corollary 3.16 Let V be a simple Lie superalgebra over k. Then

Aut_{k−conf}(Curr (V)) = Aut_{k−Lie}(V),

where Aut_{k−Lie}(V) is the group of Lie superalgebra automorphisms of V.

Proof Lie automorphisms of V extend uniquely to superconformal automorphisms of Curr (V) by k[∂]-linearity. Conversely, superconformal automorphisms of Curr (V) restrict to automorphisms of the Lie superalgebra V by Theorem 3.4. These correspondences are clearly inverse to one another. □

Corollary 3.17 Let V be a Lie superalgebra over k, and let S be an extension in k − δalg for which V ⊗ S is semisimple as a Lie superalgebra.^{8} Then

Aut_{S−conf}(Curr (V) ⊗_{k} S) = Aut_{S−Lie}(V ⊗_{k} S).

Proof Every S-conformal automorphism σ of Curr (V) ⊗_{k} S is also a kconformal automorphism via the restriction functor. By Theorem 3.4,

σ(V ⊗ S) ⊆ V ⊗ S.

The λ-bracket in the S-conformal superalgebra Curr (V) ⊗_{k} S is

[v ⊗ r_{λ}w ⊗ s] = [v, w] ⊗ rs

for all v, w ∈ V and r, s ∈ S. That is, Curr (V) ⊗_{k} S = Curr (V ⊗_{k} S) as S-conformal superalgebras. Then the argument of Corollary 3.16 holds in the S-conformal context as well. □

Remark 3.18 Let V be a finite-dimensional simple Lie superalgebra over k. By Theorem 2.16 and Proposition 2.29, the R-isomorphism classes of Ŝ/R-forms of the R-conformal superalgebra

Curr (V) ⊗_{k} R = (k[∂] ⊗_{k} V ) ⊗_{k} R

8For example, any finite-dimensional simple Lie algebra V over k satisfies this condition with any extension S ∈ k − δalg.

are parametrized by

H^{1}(Ŝ/R, Aut(Curr (V))) ≃ H_{ct}^{1}(π_{1}(R), Aut(Curr (V))(Ŝ)),

By Corollary 3.17, Aut(Curr (V))(Ŝ) = Aut_{Ŝ−Lie}(V ⊗ Ŝ), a group that is computed in [7, 13, 15]. For example, if V = 𝔰𝔩_{2}(k), then Aut_{Ŝ−Lie}(V ⊗Ŝ) = PGL_{2}(Ŝ), so

H^{1}(S/R, Aut(Curr (V)) = H_{ct}^{1}(π_{1}(R), PGL_{2}(Ŝ)) = H_{ét}^{1}(R, PGL_{2}) = {1}.

In particular, all Ŝ/R-forms of Curr (𝔰𝔩_{2}(k)) ⊗_{k} R are trivial, that is, isomorphic to Curr (𝔰𝔩_{2}(k)) ⊗_{k} R as an R-conformal superalgebra.

3.2 The Virasoro conformal algebra

The (centreless) Virasoro conformal algebra Vir is the free k[∂]-module

Vir = k[∂] ⊗ kL,

with λ-bracket given by

[L_{λ}L] = (∂ + 2λ)L.

Proposition 3.19 Let V̂ir = Vir ⊗_{k} Ŝ. Then Aut_{Ŝ−conf}V̂ir = 1.

Proof Let σ ∈ Aut_{Ŝ−conf}V̂ir, and write σ(L ⊗ 1) = ∑_{j=0}^{N} ˆ∂^{(j)}(L ⊗ r_{j}) for

some r_{j} ∈ Ŝ, where ˆ∂ = ∂V̂ir and r_{N} ≠ 0. Comparing the coefficients of the highest powers of λ occurring in the relation

σ [L ⊗ 1_{λ}L ⊗ 1] = [σ(L ⊗ 1)_{λ}σ(L ⊗ 1)] ,  (3.20)

we see that σ(L ⊗ 1) = L ⊗ r for some r ∈ Ŝ. Applying (3.20) again, we obtain r = r^{2}. Since σ is an automorphism, it is nonzero and Ŝ-linear, so r = 1 and σ = 1. □

In particular, all Ŝ/R-forms of Vir ⊗_{k} R are R-isomorphic (and hence k-isomorphic), since

H^{1}(Ŝ/R, Aut(Vir)) = H_{ct}^{1}(π_{1}(R), 1) = {1}.

It is straightforward to check that not only is V̂ir centreless^{9}, but V̂ir has no abelian conformal subalgebras:

9Hence it satisfies the hypotheses of Cor 2.39, classifying Ŝ/R-forms of Vir ⊗_{k} R up to k-isomorphism of universal central extensions.

Lemma 3.21 If A ⊆ V̂ir is a conformal subalgebra, then [A_{λ}A] ≠ 0.

Proof Let u = ∑_{j=0}^{N} ˆ∂^{(j)}(L ⊗ r_{j}) be an element of a conformal subalgebra

A ⊆ V̂ir, with r_{N} ≠ 0. Then

[u_{λ}u] = ∑_{0≤i,j≤N}(−λ)^{(i)}(∂ + λ)^{(j)} [L ⊗ r_{i}_{λ}L ⊗ r_{j}] ,

so u_{(2N+1)}u = \frac{2(−1)^{N}}{(N!)^{2}} L ⊗ r_{N}^{2} ≠ 0. □

The above lemma is in stark contrast to the fact that every current conformal algebra is linearly spanned by abelian conformal subalgebras:

Curr 𝔤 = ∑_{x∈𝔤}k[∂] ⊗ kx

and the subalgebra k[∂] ⊗ kx = Curr kx has trivial λ-bracket. Since V̂ir has no abelian conformal subalgebras, any homomorphism Curr 𝔤 → V̂ir must be trivial.

Corollary 3.22 Let 𝔤 be a Lie algebra over k, and let σ : Curr 𝔤 → V̂ir be a homomorphism of k-conformal algebras. Then σ = 0.

□

3.3 Forms of the N = 2 conformal superalgebra

Recall that the classical N = 2 k-conformal superalgebra A is the free k[∂]- module A = k[∂] ⊗_{k} V where V = V\overline{0} ⊕ V\overline{1},

\begin{aligned}V\overline{0} & = kL ⊕ kJ \\ V\overline{1} & = kG^{+} ⊕ kG^{−}, \end{aligned}

with λ-bracket given by^{10}

10These conditions say that J (respectively, G^{±}) is a primary eigenvector of conformal weight 1 (resp., \frac{3}{2} ) with respect to the Virasoro element L.

[L_{λ}L] = (∂ + 2λ)L  (3.23)
[L_{λ}J] = (∂ + λ)J  (3.24)
[L_{λ}G^{±}] = (∂ + \frac{3}{2}λ)G^{±}  (3.25)
[J_{λ}J] = 0  (3.26)
[J_{λ}G^{±}] = ±G^{±}  (3.27)
[G^{+}_{λ}G^{+}] = [G^{−}_{λ}G^{−}] = 0  (3.28)
[G^{+}_{λ}G^{−}] = L + \frac{1}{2}(∂ + 2λ)J.  (3.29)

Proposition 3.30 Let Â = A ⊗_{k} Ŝ. Then

(1) For each s = αt^{q} ∈ Ŝ^{×}, with α ∈ k^{×} and q ∈ ℚ, there exists a unique automorphism θ_{s} ∈ AutŜ(Â) such that

θ_{s} : L ↦ L + qJ ⊗ t^{−1}
J ↦ J
G^{+} ↦ G^{+} ⊗ s
G^{−} ↦ G^{−} ⊗ s^{−1}.

(2) There exists a unique automorphism ω ∈ AutŜ(Â) such that

ω : L ↦ L
J ↦ −J
G^{+} ↦ G^{−}
G^{−} ↦ G^{+}.

(3) The map s ↦ θ_{s} is a group isomorphism between Ŝ^{×} and the subgroup ⟨θ_{s}⟩_{s∈Ŝ^{×}} of AutŜ(A ⊗_{k} Ŝ) generated by the θ_{s}. This isomorphism is compatible with the action of the algebraic fundamental group π_{1}(R).

(4) Let ℤ/2ℤ act on Ŝ^{×} by \overline{1}t^{p/q} = t^{−p/q}. There exists an isomorphism of

π_{1}(R)-groups

ψ : Ŝ^{×} ⋊ ℤ/2ℤ → AutŜ(A ⊗_{k} Ŝ)

such that

ψ(s, \overline{ε}) ↦ θ_{s}ω^{ε}

for all s ∈ Ŝ^{×} and ε = 0, 1.

Proof The proofs of (1) and (2) are based on Lemma 3.1. One must check that θ_{s} and ω preserve the λ-product of any two elements of V ⊗ 1. This is done by direct (tedious) calculations.

(3) This is a straightforward consequence of the various definitions.

(4) The delicate point is to show that the (θ_{s})_{s∈Ŝ^{×}} and ω generate AutŜ(A⊗_{k} Ŝ). To do this we start with an arbitrary element σ ∈ AutŜ(A⊗_{k} Ŝ) and reason as follows:

Step 1: σ(J ⊗ 1) = J ⊗ s for some s ∈ Ŝ. Proof: Let π : Â\overline{0} → V̂ir be the projection onto the second component of

Â\overline{0} = ((k[∂] ⊗_{k} kJ) ⊗_{k} Ŝ) ⊕ ((k[∂] ⊗_{k} kL) ⊗_{k} Ŝ) = Curr (kJ ⊗_{k} Ŝ) ⊕ V̂ir.

The map πσ : Â\overline{0} → V̂ir restricts to a map

Curr (kJ) = (k[∂] ⊗ kJ) ⊗ 1 → V̂ir

of k-conformal algebras. By Corollary 3.22, this map is zero, and

σ(J ⊗ 1) = ∑_{k=0}^{N} ∂_{A⊗Ŝ}^{(k)}(J ⊗ s_{k}),

for some s_{k} ∈ Ŝ.

Comparing the coefficients of the highest powers of λ occurring in the relation

λσ(J ⊗ 1) = σ [J ⊗ 1_{λ}L ⊗ 1] = [σ(J ⊗ 1)_{λ}σ(L ⊗ 1)]

shows that N = 0. Hence σ(J ⊗ 1) = J ⊗ s for some s ∈ Ŝ.

Step 2: There exist c ∈ k^{×} and s ∈ Ŝ such that σ(J ⊗ 1) = J ⊗ c and σ(L ⊗ 1) = L ⊗ 1 + J ⊗ s. Proof: Apply σ to the relation [J ⊗ 1_{λ}L ⊗ 1] = λJ ⊗ 1. Comparing the coefficients of the top powers of λ shows that σ(J ⊗ 1) = J ⊗ c for some c ∈ k^{×} and

σ(L ⊗ 1) = L ⊗ 1 + ∑_{j=0}^{N} ˆ∂^{(j)}(J ⊗ s_{j}),

for some s_{j} ∈ Ŝ, where s_{N} ≠ 0 and ˆ∂ = ∂_{A⊗Ŝ}. We then consider the relation

[σ(L ⊗ 1)_{λ}σ(L ⊗ 1)] = (ˆ∂ + 2λ)σ(L ⊗ 1).  (3.31)

Expanding and comparing the coefficients of λˆ∂^{(N)}(J ⊗ s_{N}) shows that N is 0 or 1.

If N = 1, then (3.31) becomes λJ ⊗s_{0} = −(ˆ∂+2λ)J ⊗δŜ(s_{1}). Since there are no ˆ∂ terms in the left-hand side of this equation, we see that δŜ(s_{1}) = 0. But then s_{0} = 0. Thus σ(L ⊗ 1) = L ⊗ 1 + ˆ∂(J ⊗ c_{1}) for some c_{1} ∈ k or else σ(L ⊗ 1) = L ⊗ 1 + J ⊗ s for some s ∈ Ŝ.

Suppose that σ(L ⊗ 1) = L ⊗ 1 + ˆ∂(J ⊗ c_{1}). Write

σ(G^{+} ⊗ 1) = ∑_{k=0}^{M_{+}} ˆ∂^{(k)}(G^{+} ⊗ r_{k}) + ∑_{ℓ=0}^{M_{−}} ˆ∂^{(ℓ)}(G^{−} ⊗ s_{ℓ}),

with r_{M_{+}} and s_{M_{−}} both nonzero. Consider the relation

σ [L ⊗ 1_{λ}G^{±} ⊗ 1] = [σ(L ⊗ 1_{λ}σ(G^{±} ⊗ 1)] .  (3.32)

Expanding this relation and comparing the coefficients of terms involving λ^{1+M_{+}}G^{+}, we see that \frac{3}{2}r_{0}λ = \frac{λ^{1+M_{+}}}{M_{+}!} r_{M_{+}} (\frac{3}{2} − c_{1}) . If r_{0} ≠ 0, then M_{+} = c_{1} = 0. Similar arguments show that if we write

σ(G^{−} ⊗ 1) = ∑_{k=0}^{N_{+}} ˆ∂^{(k)}(G^{+} ⊗ a_{k}) + ∑_{ℓ=0}^{N_{−}} ˆ∂^{(ℓ)}(G^{−} ⊗ b_{ℓ}),

then c_{1} = 0 if a_{0} ≠ 0. But if r_{0} = a_{0} = 0, then the map σ is not surjective, so c_{1} = 0 and σ(L ⊗ 1) = L ⊗ 1 + J ⊗ s for some s ∈ Ŝ.

Step 3: There exist s^{+} and s^{−} in Ŝ^{×} such that either 3(a) σ(J ⊗ 1) = J ⊗ 1 and σ(G^{±} ⊗ 1) = G^{±} ⊗ s^{±}, or 3(b) σ(J ⊗ 1) = −J ⊗ 1 and σ(G^{±} ⊗ 1) = G^{∓} ⊗ s^{±}.

Proof: From the proof of Step 2, we see that M_{+} = 0. Likewise, M_{−} = N_{+} = N_{−} = 0. Hence

σ(G^{+} ⊗ 1) = G^{+} ⊗ r_{+} + G^{−} ⊗ s_{+}
σ(G^{−} ⊗ 1) = G^{+} ⊗ r_{−} + G^{−} ⊗ s_{−},

for some r_{±}, s_{±} ∈ Ŝ. The relation σ [J ⊗ 1_{λ}G^{±} ⊗ 1] = [σ(J ⊗ 1)_{λ}σ(G^{±} ⊗ 1)] shows that

G^{+} ⊗ r_{+} + G^{−} ⊗ s_{+} = cG^{+} ⊗ r_{+} − cG^{−} ⊗ s_{+},  (3.33)
−(G^{+} ⊗ r_{−} + G^{−} ⊗ s_{−}) = cG^{+} ⊗ r_{−} − cG^{−} ⊗ s_{−}.  (3.34)

If r_{+} ≠ 0, then by (3.33), we see that s_{+} = 0 and c = 1. Then by (3.34), r_{−} = 0 and s_{−} ≠ 0. Hence we are in Case 3(a).

Similarly, if r_{+} = 0, then s_{+} ≠ 0, c = −1, and we are in Case 3(b).

Step 4: If σ is as in Case 3(a) above, then s^{+} and s^{−} are inverses of each other. Furthermore, if we write s^{+} = αt^{q} for some α ∈ k^{×} and q ∈ ℚ, then σ = θ_{s^{+}}. Proof: Since we are in Case 3(a), the relation

σ [G^{+} ⊗ 1_{λ}G^{−} ⊗ 1] = [σ(G^{+} ⊗ 1)_{λ}σ(G^{−} ⊗ 1)]

says that

L ⊗ 1 + J ⊗ s + \frac{1}{2}(ˆ∂ + 2λ)(J ⊗ 1)

= L ⊗ s^{+}s^{−} + J ⊗ δŜ(s^{+})s^{−} + \frac{1}{2}∂J ⊗ s^{+}s^{−} + λJ ⊗ s^{+}s^{−}.

Thus s^{+} and s^{−} are inverses of each other, and

s = δŜ(s^{+})s^{−} = qαt^{q−1}(s^{+})^{−1} = qt^{−1}.

Hence σ = θ_{s^{+}}.

To finish the proof of Proposition 3.30, we have to consider the case when σ is as in 3(b). Replacing σ by σω yields an automorphism that satisfies 3(a), and we can conclude by Step 4. □

Theorem 3.35 Let A be the classical N = 2 conformal superalgebra. Up to k-conformal isomorphism, there are exactly two twisted loop algebras of A. These are L(A, id) and L(A, ω). Furthermore, any Ŝ/R-form of A is isomorphic to one of these two loop algebras.

Proof By Theorem 2.16, Proposition 2.29, and Proposition 3.30, the Risomorphism classes of Ŝ/R-forms of the R-conformal superalgebra A are parametrized by

H^{1}(Ŝ/R, Aut(A)) ≃ H_{ct}^{1}(π_{1}(R), Aut(A)(Ŝ)) ≃ H_{ct}^{1}(π_{1}(R), Ŝ^{×} ⋊ ℤ/2ℤ)).

Consider the split exact sequence of π_{1}(R) = ℤ̂-groups

1 → Ŝ^{×} → Ŝ^{×} ⋊ ℤ/2ℤ → ℤ/2ℤ → 1.  (3.36)

Passing to (continuous) cohomology yields

H_{ct}^{1}(ℤ̂, Ŝ^{×}) → H_{ct}^{1}(ℤ̂, Ŝ^{×} ⋊ ℤ/2ℤ) →^{ψ} H_{ct}^{1}(ℤ̂, ℤ/2ℤ).  (3.37)

The map ψ admits a section (hence is surjective) because the sequence (3.36) is split. Since ℤ̂ acts trivially on the (abelian) group ℤ/2ℤ, we have

H_{ct}^{1}(ℤ̂, ℤ/2ℤ) ≃ Hom_{ct}(ℤ̂, ℤ/2ℤ) ≃ ℤ/2ℤ.

Note that ψ maps the cohomology classes of H_{ct}^{1}(ℤ̂, AutŜ(A ⊗_{k} Ŝ)) corresponding to the loop algebras L(A, id) and L(A, ω) to the two distinct classes of H_{ct}^{1}(ℤ̂, ℤ/2ℤ). To prove Theorem 3.35, it is thus enough to show that ψ in (3.37) is bijective, and that Ŝ/R-forms of A are k-conformal isomorphic if and only if they are R-conformal isomorphic.

Let G_{m} = Spec(k[z^{±1}]) denote the multiplicative group. Recall that Aut(G_{m}) ≃ ℤ/2ℤ where the generator \overline{1} of ℤ/2ℤ acts on Spec(k[z^{±1}]) via z ↦ z^{−1}. We now proceed by exploiting the considerations of Remark 2.30. Since H_{ét}^{1}(R, ℤ/2ℤ) ≃ ℤ/2ℤ, the bijectivity of ψ translates into the bijectivity of the analoguous map (also denoted by ψ) at the étale level, namely

H_{ét}^{1}(R, G_{m}) → H_{ét}^{1}(R, G_{m} ⋊ ℤ/2ℤ) →^{ψ} H_{ét}^{1}(R, ℤ/2ℤ)

Since H_{ét}^{1}(R, G_{m}) = Pic (R) = 1 our map ψ has trivial kernel. The fibre of ψ over the non–trivial class of H^{1}(R, ℤ/2ℤ) is measured by the cohomology H_{ét}^{1}(R, R_{S_{2}/R}^{1}(G_{m})) where R_{S_{2}/R}^{1}(G_{m}) is the twisted form of the multiplicative R-group G_{m} that fits into the exact sequence

1 ⟶ R_{S_{2}/R}^{1}(G_{m}) ⟶ R_{S_{2}/R}(G_{m}) \underrightarrow{N̂} G_{m} ⟶ 1,

where N̂ comes from the reduced norm N of the quadratic extension S_{2}/R, and R is the Weil restriction. The functor of points of the R-group R_{S_{2}/R}^{1}(G_{m}) is thus given by

R_{S_{2}/R}^{1}(G_{m})(R′) = {x ∈ (S_{2} ⊗_{R} R′)^{×} : N(x) = 1}.

for all R′ ∈ R−alg. Passing to cohomology on this last exact sequence yields

R_{S_{2}/R}^{1}(G_{m})(R) →^{N} G_{m}(R) → H^{1}(R, R_{S_{2}/R}^{1}(G_{m})) → H^{1}(R, R_{S_{2}/S}(G_{m})).

By Shapiro’s Lemma,

H^{1}(R, R_{S_{2}/S}(G_{m})) = Pic (S) = 0.

On the other hand, the norm map

{x ∈ S_{2}^{×} : N(x) = 1} ⟶^{N} R^{×}

is surjective. Thus H^{1}(R, R_{S_{2}/R}^{1}(G_{m})) = 1 as desired.^{11}

The above cohomological reasoning shows that L(A, id) and L(A, ω) are nonisomorphic as R-conformal superalgebras, and they represent the Risomorphism classes of Ŝ/R-forms of A. To finish the proof, it suffices to note that the hypotheses of Corollary 2.36 are satisfied, so (in this case) R-isomorphism is the same as k-isomorphism. This follows easily from the same argument used in the N = 4 case in the proof of Theorem 3.71 below. □

3.4 Forms of the N = 4 conformal superalgebra

We now consider the classical N = 4 conformal superalgebra A. We compute the automorphism group Aut(A)(Ŝ), from which the existence of infinitely many non-isomorphic twisted loop algebras will follow easily from the theory developed in §2.2.

We begin by recalling the definition of the N = 4 conformal superalgebra A. Let A = A\overline{0} ⊕ A\overline{1} be the k-vector space with

A\overline{ι} = k[∂]V\overline{ι} ≅ V\overline{ι} ⊗_{k} k[∂]  (3.38)

for \overline{ι} = \overline{0}, \overline{1}, and

V\overline{0} = kL ⊕ ⨁_{s=1}^{3} kJ^{s}

V\overline{1} = ⨁_{a=1}^{2} kG^{a} ⊕ ⨁_{b=1}^{2} k\overline{G}^{b}.

11One can also see that H^{1}(R, R_{S_{2}/R}^{1}(G_{m})) = 1 directly by applying Remark 2.30 (2)

to the reductive R-group R_{S_{2}/R}^{1}(G_{m}).

Let J^{s} = \frac{1}{2}σ^{s}, where σ^{s} are the Pauli spin matrices

σ^{1} = ( \begin{matrix}0 & 1 \\ 1 & 0 \end{matrix} )
σ^{2} = ( \begin{matrix}0 & −i \\ i & 0 \end{matrix} )
σ^{3} = ( \begin{matrix}1 & 0 \\ 0 & −1 \end{matrix} ) .

The space A is a (k, δ)-conformal superalgebra, with multiplication given by

[L_{λ}L] = (∂ + 2λ)L  (3.39)
[L_{λ}J^{s}] = (∂ + λ)J^{s}  (3.40)
[J^{m}_{λ}J^{n}] = [J^{m}, J^{n}] := J^{m}J^{n} − J^{n}J^{m}  (3.41)
[L_{λ}G^{a}] = (∂ + \frac{3}{2}λ)G^{a}  (3.42)
[L_{λ}\overline{G}^{a}] = (∂ + \frac{3}{2}λ)\overline{G}^{a}  (3.43)
[J^{s}_{λ}G^{a}] = −\frac{1}{2} ∑_{b=1}^{3} σ_{ab}^{s}G^{b}  (3.44)

[J^{s}_{λ}\overline{G}^{a}] = \frac{1}{2} ∑_{b=1}^{3} σ_{ba}^{s}\overline{G}^{b}  (3.45)
[G^{a}_{λ}G^{b}] = [\overline{G}^{a}_{λ}\overline{G}^{b}] = 0  (3.46)
[G^{a}_{λ}\overline{G}^{b}] = 2δ_{ab}L − 2(∂ + 2λ) ∑_{s=1}^{3} σ_{ab}^{s}J^{s}  (3.47)

for all m, n, s ∈ {1, 2, 3} and a, b ∈ {1, 2}, where δ_{ab} is the Kronecker delta and σ_{ab}^{s} is the (a, b)-entry of the matrix σ^{s}. The algebra A is the N = 4 conformal superalgebra described in [9].

To compute Aut(A)(Ŝ) = AutŜ(A⊗_{k} Ŝ), it is enough (by Lemma 3.1) to compute the action of each automorphism in AutŜ(A ⊗_{k} Ŝ) on the subspace V ⊗_{k} 1 ⊆ A ⊗_{k} Ŝ. Fix σ ∈ AutŜ(A ⊗_{k} Ŝ), and choose d > 0 sufficiently large so that σ(V ⊗ 1) ⊆ A_{d} := A ⊗ k[t^{1/d}, t^{−1/d}]. (Such a d exists since V is finite-dimensional.) Let z = t^{1/d} and δ̂ = \frac{d}{dt} : S_{d} → S_{d} with S_{d} = k[t^{1/d}, t^{−1/d}] = k[z, z^{−1}].

As in the N = 2 case, we consider the projection

π : (A ⊗ Ŝ)\overline{0} = Curr 𝔰𝔩_{2}(Ŝ) ⊕ V̂ir ⟶ V̂ir

and the composite map

πσ : (A ⊗ Ŝ)\overline{0} ⟶ V̂ir

restricted to Curr 𝔰𝔩_{2}(k) = (k[∂] ⊗ ⨁_{s=1}^{3} kJ^{s}) ⊗ 1.

By Corollary 3.22, the map πσ is zero, and thus σ restricts to an Ŝconformal automorphism of the conformal subalgebra Curr 𝔰𝔩_{2}(Ŝ) = (k[∂] ⊗ 𝔰𝔩_{2}(k) ⊗_{k} Ŝ ⊆ A ⊗ Ŝ. By Corollary 3.17, σ is the k[∂]-linear extension of an Ŝ-Lie algebra automorphism of 𝔰𝔩_{2}(Ŝ), so by [13], there is some Y ∈ GL_{2}(Ŝ) such that

σ(J^{s} ⊗ 1) = Y J^{s}Y^{−1}  (3.48)

for s = 1, 2, 3. The units of Ŝ are the monomials, so det Y = ct^{q} for some q ∈ ℚ and nonzero c ∈ k. Since k is assumed to be algebraically closed, c has a square root \sqrt{c} ∈ k. Let Ŷ = \frac{1}{\sqrt{c}}t^{−q/2}Y. Then Ŷ has determinant 1,

and conjugation by Ŷ has the same effect on J^{s} as conjugation by Y, so we can assume without loss of generality that Y ∈ SL_{2}(Ŝ).

Next we consider the image of L ⊗ 1. Write

σ(L ⊗ 1) = ∑_{i∈ℤ}P_{i}(ˆ∂)(L ⊗ z^{i}) + w

for some polynomials P_{i}(ˆ∂) in the polynomial ring k[ˆ∂] and w ∈ Curr 𝔰𝔩_{2}(Ŝ). Note that {i ∈ ℤ | P_{i} ≠ 0} is nonempty (or else σ would not be surjective). Set N = max{i ∈ ℤ | P_{i} ≠ 0} and M = min{i ∈ ℤ | P_{i} ≠ 0}. If N > 0, then comparing the coefficients of L ⊗ z^{2N} on both sides of

[σ(L ⊗ 1)_{λ}σ(L ⊗ 1)] = σ [L ⊗ 1_{λ}L ⊗ 1]  (3.49)

gives

P_{N}(−λ)P_{N}(ˆ∂ + λ)(ˆ∂ + 2λ)(L ⊗ z^{2N}) = 0.

That is, P_{N} = 0, a contradiction. Hence N ≤ 0.

If N < 0, then comparing the coefficients of L ⊗ z^{2M−d} in (3.49) gives

P_{M}(−λ)P_{M}(ˆ∂ + λ)(−M/d)(L ⊗ z^{2M−d}) = 0,

keeping in mind that ∂L ⊗ z^{NM} = ˆ∂(L ⊗ z^{M}) − L ⊗ δ̂(z^{M}) with δ̂ = \frac{d}{dt} and z = t^{1/d}. Hence P_{M} = 0, another contradiction. Thus N = M = 0.

Let P (∂) := P_{0}(∂). Then comparing the coefficients of L ⊗ 1 in (3.49) gives

P (−λ)P (ˆ∂ + λ)(ˆ∂ + 2λ)(L ⊗ 1) = P (ˆ∂)(ˆ∂ + 2λ)(L ⊗ 1),

so P (∂) is a constant (i.e. a member of k) and P^{2} = P. Since {i ∈ ℤ | P_{i} ≠ 0} is nonempty, P ≠ 0, so P = 1. Hence

σ(L ⊗ 1) = L ⊗ 1 + w

for some w ∈ Curr 𝔰𝔩_{2}(Ŝ).

Write w = ∑_{j=0}^{N′} ˆ∂^{(j)}(w_{j}) for some w_{j} ∈ 𝔰𝔩_{2}(Ŝ), with w_{N′} ≠ 0. Suppose N′ > 0. For u ∈ 𝔰𝔩_{2}(Ŝ) = ⨁_{s=1}^{3} kJ^{s} ⊗ Ŝ,

[u_{λ}L ⊗ 1] = (1 ⊗ δ̂)u + λu,

so

σ((1 ⊗ δ̂)u) + λσ(u) = [σ(u)_{λ}σ(L ⊗ 1)]
= (1 ⊗ δ̂)σ(u) + λσ(u) + ∑_{j=0}^{N′}(ˆ∂ + λ)^{(j)}[gs(u), w_{j}],

using the fact that σ(u) ∈ 𝔰𝔩_{2}(Ŝ) (Corollary 3.17). Comparing powers of ˆ∂ gives [σ(u), w_{N′}] = 0. This holds for all u ∈ 𝔰𝔩_{2}(Ŝ), so w_{N′} is in the centre Z(𝔰𝔩_{2}(Ŝ)) = 0 of the Lie algebra 𝔰𝔩_{2}(Ŝ). Therefore w_{N′} = 0, a contradiction. Hence w ∈ 𝔰𝔩_{2}(Ŝ).

Hence we have now proven the following lemma:

Lemma 3.50 Let A be the N = 4 conformal superalgebra defined above. Then for any σ ∈ Aut_{Ŝ−conf}(A ⊗ Ŝ), there is some Y ∈ SL_{2}(Ŝ) and some w ∈ 𝔰𝔩_{2}(Ŝ) so that

σ(J^{s} ⊗ 1) = Y J^{s}Y^{−1}
σ(L ⊗ 1) = L ⊗ 1 + w

for all s ∈ {1, 2, 3}. □

Our next task is to find the value of w in Lemma 3.50. For all u ∈ 𝔰𝔩_{2}(Ŝ), we have

(ˆ∂ + λ)σ^{−1}(u) − \frac{d}{dt}σ^{−1}(u) = [L ⊗ 1_{λ}σ^{−1}(u)] ,

so

(ˆ∂ + λ)u − σ(\frac{d}{dt}σ^{−1}(u)) = [σ(L ⊗ 1)_{λ}u]
= [L ⊗ 1 + w_{λ}u]
= (ˆ∂ + λ)u − \frac{d}{dt}u + [w, u],

and

[w, u] = \frac{d}{dt}u − σ(\frac{d}{dt}σ^{−1}(u))
= u′ − Y (Y^{−1}uY)′Y^{−1},

where prime (′) denotes the derivative taken with respect to the variable t. But

Y′Y^{−1} + Y (Y^{−1})′ = (Y Y^{−1})′ = 0,  (3.51)

so

[w, u] = u′ − Y (Y^{−1}uY)′Y^{−1}
= −Y (Y^{−1})′u − uY′Y^{−1}
= [Y′Y^{−1}, u].

Thus w − Y′Y^{−1} is in the centralizer of 𝔰𝔩_{2}(Ŝ) in 𝔤𝔩_{2}(Ŝ). But writing

Y = ( \begin{matrix}e & f \\ g & h \end{matrix} ) ,

with e, f, g, h ∈ Ŝ, a quick computation shows that the trace tr(Y′Y^{−1}) = (eh − fg)′. But (eh − fg)′ = 0 since Y ∈ SL_{2}(Ŝ). Hence, w − Y′Y^{−1} ∈ Z(𝔰𝔩_{2}(Ŝ)) = 0, the centre of 𝔰𝔩_{2}(Ŝ), so we have the following proposition.

Proposition 3.52 Let A be the N = 4 conformal superalgebra defined above. Then for any σ ∈ Aut_{Ŝ−conf}(A ⊗ Ŝ), there is some Y ∈ SL_{2}(Ŝ) so that

σ(J^{s} ⊗ 1) = Y J^{s}Y^{−1}
σ(L ⊗ 1) = L ⊗ 1 + Y′Y^{−1}

for all s ∈ {1, 2, 3}. □

Next we consider the action of σ on the odd part of A ⊗ Ŝ. Let v ∈ V\overline{1}. Write

σ(v ⊗ 1) = ∑_{i=0}^{M′} ∑_{s=1}^{4} ˆ∂^{(i)}(v_{s} ⊗ r_{si}),

where v_{1} = G^{1}, v_{2} = G^{2}, v_{3} = \overline{G}^{1}, v_{4} = \overline{G}^{2}, and r_{sM′} ≠ 0 for some s. Then writing w for Y′Y^{−1}, we have

(ˆ∂ + \frac{3}{2}λ)σ(v ⊗ 1) = σ [L ⊗ 1_{λ}v ⊗ 1]
= [σ(L ⊗ 1)_{λ}σ(v ⊗ 1)]
= ∑_{i=0}^{M′} ∑_{s=1}^{4}(ˆ∂ + λ)^{(i)}((ˆ∂ + \frac{3}{2}λ)(v_{s} ⊗ r_{si})
− v_{s} ⊗ δ̂(r_{si}) + [w_{λ}v_{s} ⊗ r_{si}] ).

From the definition of the relevant products, (3.42), (3.43), and (1.13), it is clear that [w_{λ}v_{s} ⊗ r_{si}] contains no nonzero powers of λ. If M′ > 0, then comparing the coefficients of λ^{M′+1} gives

0 = ∑_{s=1}^{4} \frac{3}{2}(M′ + 1)λ^{(M′+1)}v_{s} ⊗ r_{sM′}.

Thus r_{sM′} = 0 for all s, a contradiction. Hence M′ ≤ 0, and σ(v⊗1) ∈ V ⊗S. Therefore, by the Ŝ-linearity of σ and Proposition 3.52, we have proven the following lemma.

Lemma 3.53 Let σ ∈ AutŜ(A ⊗ Ŝ). Then σ(V ⊗ Ŝ) ⊆ V ⊗ Ŝ. □

For the computations that follow, it will be helpful to use the following notation:

( \begin{matrix}a \\ b \end{matrix} ) ⊗ ( \begin{matrix}1 \\ 0 \end{matrix} ) := G^{1} ⊗ a + G^{2} ⊗ b

( \begin{matrix}a \\ b \end{matrix} ) ⊗ ( \begin{matrix}0 \\ 1 \end{matrix} ) := \overline{G}^{1} ⊗ a + \overline{G}^{2} ⊗ b

for all a, b ∈ Ŝ. In this notation, the relations (3.44), (3.45), and (3.47) become

[J^{s}_{λ}( \begin{matrix}a \\ b \end{matrix} ) ⊗ ( \begin{matrix}1 \\ 0 \end{matrix} )] = −(J^{s})^{T} ( \begin{matrix}a \\ b \end{matrix} ) ⊗ ( \begin{matrix}1 \\ 0 \end{matrix} )  (3.54)
[J^{s}_{λ}( \begin{matrix}g \\ h \end{matrix} ) ⊗ ( \begin{matrix}0 \\ 1 \end{matrix} )] = J^{s} ( \begin{matrix}g \\ h \end{matrix} ) ⊗ ( \begin{matrix}0 \\ 1 \end{matrix} )  (3.55)

[( \begin{matrix}a \\ b \end{matrix} ) ⊗ ( \begin{matrix}1 \\ 0 \end{matrix} )_{λ}( \begin{matrix}g \\ h \end{matrix} ) ⊗ ( \begin{matrix}0 \\ 1 \end{matrix} )]

= 2 ( a b ) ( \begin{matrix}g \\ h \end{matrix} ) L − 2(∂ + 2λ) ∑_{s=1}^{3} ( a b ) σ^{s} ( \begin{matrix}g \\ h \end{matrix} ) J^{s}. (3.56)

for all a, b, g, h ∈ k and s ∈ {1, 2, 3}, where (J^{s})^{T} is the transpose of the matrix J^{s}.

Using the fact that σ preserves the relation (3.54), we see that for any a, b ∈ k,

−(Y J^{s}Y^{−1})^{T} σ_{1} (( \begin{matrix}a \\ b \end{matrix} ) ⊗ ( \begin{matrix}1 \\ 0 \end{matrix} )) + Y J^{s}Y^{−1}σ_{2} (( \begin{matrix}a \\ b \end{matrix} ) ⊗ ( \begin{matrix}1 \\ 0 \end{matrix} ))
= −σ_{1} ((J^{s})^{T} ( \begin{matrix}a \\ b \end{matrix} ) ⊗ ( \begin{matrix}1 \\ 0 \end{matrix} )) − σ_{2} ((J^{s})^{T} ( \begin{matrix}a \\ b \end{matrix} ) ⊗ ( \begin{matrix}1 \\ 0 \end{matrix} )) ,

where σ_{i} := π_{i}σ and π_{1} (resp., π_{2}) is the projection of A\overline{1} ⊗ Ŝ onto

W_{1} := ⨁_{j=1}^{2} kG^{j} ⊗ Ŝ (resp., W_{2} := ⨁_{j=1}^{2} k\overline{G}^{j} ⊗ Ŝ)

in the direct sum

A\overline{1} = W_{1} ⊕ W_{2}.

Then

−(Y J^{s}Y^{−1})^{T} σ_{1} = −σ_{1}(J^{s})^{T}  (3.57)

as Ŝ-linear maps on the vector space W_{1}. Let v = ( \begin{matrix}v_{1} \\ v_{2} \end{matrix} ) ⊗ ( \begin{matrix}1 \\ 0 \end{matrix} ) be in

the kernel ker σ_{1} of the restriction σ_{1} : W_{1} → W_{1}. Then by (3.57),

Mv := (M ( \begin{matrix}v_{1} \\ v_{2} \end{matrix} )) ⊗ ( \begin{matrix}1 \\ 0 \end{matrix} ) ∈ ker σ_{1}

for all M ∈ 𝔰𝔩_{2}(Ŝ). Thus ker σ_{1} = 0 or ker σ_{1} = W_{1}. If ker σ_{1} = 0, we see by (3.57) that conjugation by σ_{1} has the same effect on 𝔰𝔩_{2}(k) as conjugation by (Y^{−1})^{T}. The centralizer of 𝔰𝔩_{2}(k) in GL_{2}(Ŝ) consists of matrices of the form cz^{m}I, where c ∈ k^{×}, m ∈ ℤ, and I is the 2 × 2 identity matrix. Thus

σ_{1} = cz^{m}(Y^{−1})^{T} : W_{1} → W_{1}  (3.58)

for some c ∈ k^{×} and m ∈ ℤ. If ker σ_{1} = W_{1}, then obviously (3.58) also holds, with c = 0.

By a similar argument,

σ_{2} = dz^{n}Y ( \begin{matrix}0 & 1 \\ −1 & 0 \end{matrix} ) : W_{1} → W_{2}

for some d ∈ k and n ∈ ℤ. Thus

σ (( \begin{matrix}a \\ b \end{matrix} ) ⊗ ( \begin{matrix}1 \\ 0 \end{matrix} ))

= cz^{m}(Y^{−1})^{T} ( \begin{matrix}a \\ b \end{matrix} ) ⊗ ( \begin{matrix}1 \\ 0 \end{matrix} ) + dz^{n}Y ( \begin{matrix}0 & 1 \\ −1 & 0 \end{matrix} ) ( \begin{matrix}a \\ b \end{matrix} ) ⊗ ( \begin{matrix}0 \\ 1 \end{matrix} ) . (3.59)

Repeating this argument on W_{2} using relation (3.55), we see that for some fixed e, f ∈ k and k, ℓ ∈ ℚ,

σ (( \begin{matrix}g \\ h \end{matrix} ) ⊗ ( \begin{matrix}0 \\ 1 \end{matrix} ))

= ez^{k}(Y^{−1})^{T} ( \begin{matrix}0 & 1 \\ −1 & 0 \end{matrix} ) ( \begin{matrix}g \\ h \end{matrix} ) ⊗ ( \begin{matrix}1 \\ 0 \end{matrix} ) + fz^{ℓ}Y ( \begin{matrix}g \\ h \end{matrix} ) ⊗ ( \begin{matrix}0 \\ 1 \end{matrix} ) (3.60)

for all g, h ∈ k.

From (3.42), we see that

[L ⊗ 1_{λ}( \begin{matrix}a \\ b \end{matrix} ) ⊗ ( \begin{matrix}1 \\ 0 \end{matrix} )] = (ˆ∂ + \frac{3}{2}λ) ( \begin{matrix}a \\ b \end{matrix} ) ⊗ ( \begin{matrix}1 \\ 0 \end{matrix} )  (3.61)

for all a, b ∈ k. Apply σ_{1} to both sides of (3.61) and compute using (3.59). Equating the terms on both sides of the equation which are constant with respect to λ and ˆ∂ gives

(cz^{m}(Y^{−1})^{T})′ = −(Y′Y^{−1})^{T} cz^{m}(Y^{−1})^{T} ,

where prime (′) denotes element-by-element differentiation with respect to t = z^{d}. Taking the transpose of both sides and simplifying using (3.51) gives c = 0 or mz^{m−1}Y^{−1} = 0. That is, c = 0 or m = 0. If c = 0, then we can obviously assume that m = 0. Similarly, n = k = ℓ = 0.

By (3.56), the following equation holds for all a, b, g, h ∈ k:

[σ (( \begin{matrix}a \\ b \end{matrix} ) ⊗ ( \begin{matrix}1 \\ 0 \end{matrix} ))_{λ} σ (( \begin{matrix}g \\ h \end{matrix} ) ⊗ ( \begin{matrix}0 \\ 1 \end{matrix} ))]

= 2 ( a b ) ( \begin{matrix}g \\ h \end{matrix} ) σ(L ⊗ 1) − 2(ˆ∂ + 2λ)σ (∑_{s=1}^{3} ( a b ) σ^{s} ( \begin{matrix}g \\ h \end{matrix} )) J^{s}. (3.62)

Then comparing the coefficients of L ⊗ 1 on both sides of (3.62) gives

2cf ( a b ) ( \begin{matrix}g \\ h \end{matrix} ) − 2de ( g h ) ( \begin{matrix}a \\ b \end{matrix} ) = 2 ( a b ) ( \begin{matrix}g \\ h \end{matrix} ) ,

so cf − de = 1. Thus we have the following proposition:

Proposition 3.63 Let σ ∈ AutŜ(A ⊗ Ŝ). Then for some Y ∈ SL_{2}(Ŝ) and

( \begin{matrix}c & d \\ e & f \end{matrix} ) ∈ SL_{2}(k), σ satisfies the following formulas

σ(L ⊗ 1) = L ⊗ 1 + Y′Y^{−1}  (3.64)
σ(J^{s} ⊗ 1) = Y J^{s}Y^{−1}  (3.65)
σ (( \begin{matrix}a \\ b \end{matrix} ) ⊗ ( \begin{matrix}1 \\ 0 \end{matrix} )) = c(Y^{−1})^{T} ( \begin{matrix}a \\ b \end{matrix} ) ⊗ ( \begin{matrix}1 \\ 0 \end{matrix} )
+ dY ( \begin{matrix}0 & 1 \\ −1 & 0 \end{matrix} ) ( \begin{matrix}a \\ b \end{matrix} ) ⊗ ( \begin{matrix}0 \\ 1 \end{matrix} ) (3.66)
σ (( \begin{matrix}a \\ b \end{matrix} ) ⊗ ( \begin{matrix}0 \\ 1 \end{matrix} )) = e(Y^{−1})^{T} ( \begin{matrix}0 & 1 \\ −1 & 0 \end{matrix} ) ( \begin{matrix}a \\ b \end{matrix} ) ⊗ ( \begin{matrix}1 \\ 0 \end{matrix} )
+ fY ( \begin{matrix}a \\ b \end{matrix} ) ⊗ ( \begin{matrix}0 \\ 1 \end{matrix} )  (3.67)

for all a, b ∈ k and s = 1, 2, 3.

□

The converse to Proposition 3.63 is that given any Y ∈ SL_{2}(Ŝ) and

( \begin{matrix}c & d \\ e & f \end{matrix} ) ∈ SL_{2}(k), (3.64)–(3.67) defines an automorphism σ ∈ AutŜ(A ⊗

Ŝ). This follows (using Lemma 3.1) from the long and tedious verification that the Ŝ-linear map σ : A ⊗ Ŝ → A ⊗ Ŝ defined by (3.64)–(3.67) preserves the λ-bracket on the following relation:

σ [w_{1} ⊗ 1_{λ}w_{2} ⊗ 1] = [σ(w_{1} ⊗ 1)_{λ}σ(w_{2} ⊗ 1)]  (3.68)

for w_{1}, w_{2} ∈ \{L, J^{s}, G^{i}, \overline{G}^{i} | s = 1, 2, 3, i = 1, 2\}.

To determine the group structure on the set of automorphisms, fix Y ∈ SL_{2}(Ŝ) and X ∈ SL_{2}(k). Let η_{1} (resp., η_{2}) denote the automorphism determined by (Y, I) (resp., (I, X)), where I is the 2 × 2 identity matrix. Then it is straightforward to verify that

η_{2}η_{1}η_{2}^{−1} = η_{1},

so there is a group epimorphism

ϕ : SL_{2}(Ŝ) × SL_{2}(k) → AutŜ(A ⊗ Ŝ).

Finally, suppose that the automorphism determined by the pair (Y, X) ∈

SL_{2}(Ŝ) × SL_{2}(k) is in the kernel ker ϕ. Writing X = ( \begin{matrix}c & d \\ e & f \end{matrix} ),

c(Y^{−1})^{T} ( \begin{matrix}a \\ b \end{matrix} ) ⊗ ( \begin{matrix}1 \\ 0 \end{matrix} ) = ( \begin{matrix}a \\ b \end{matrix} ) ⊗ ( \begin{matrix}1 \\ 0 \end{matrix} )

for all a, b ∈ k by (3.66). Thus c(Y^{−1})^{T} = I, so Y = cI and c = ±1. By (3.66) and (3.67), we also see that d = e = 0 and Y = fI. Since (−I, −I) determines the identity map on A ⊗ Ŝ by (3.64)–(3.67), the kernel of ϕ is the subgroup of SL_{2}(Ŝ) × SL_{2}(k) generated by (−I, −I):

ker(ϕ) = ⟨(−I, −I)⟩ ≃ ℤ/2ℤ

We have now proven the following:

Proposition 3.69 Let A be the N = 4 conformal superalgebra defined above, and let Ŝ = k[t^{q} | q ∈ ℚ]. Then

Aut_{Ŝ−conf}(A ⊗_{k} Ŝ) = \frac{SL_{2}(Ŝ) × SL_{2}(k)}{⟨(−I, −I)⟩}.  (3.70)
□

Applying our theory of forms (§2) now allows us to classify twisted loop algebras of the N = 4 conformal superalgebra A.

Theorem 3.71 Let A be the N = 4 conformal superalgebra defined above. Then there are canonical bijections between the following sets:

(i) R-isomorphism classes of twisted loop algebras of A, (ii) k-isomorphism classes of twisted loop algebras of A,

(iii) R-isomorphism classes of Ŝ/R-forms of A ⊗_{k} R,

(iv) conjugacy classes of elements of finite order in PGL_{2}(k),

Proof Let A_{m} be the S_{m}-conformal superalgebra A ⊗_{k} S_{m} where S_{m} = (S_{m}, \frac{d}{dt}), S_{m} = k[t^{± \frac{1}{m}}], and m ≥ 1. Let V = V\overline{0} ⊕ V\overline{1}, where V\overline{0} and V\overline{1} are defined as in (3.38). We divide our proof that (i) and (ii) are equivalent into several steps.

Step 1: A_{m} = Span_{k}{v_{(1)}∂_{A}^{(ℓ)}L ⊗ 1 | v ∈ V ⊗ S_{m}, ℓ ≥ 0} Proof: We show that ∂_{A}^{ℓ}V ⊗ S_{m} ⊆ ∑_{j=0}^{ℓ} V ⊗ S_{m}_{(1)}∂_{A}^{(j)}L ⊗ 1 using induction on ℓ. For ℓ = 0, we see that V ⊗ S_{m}_{(1)}L ⊗ 1 = (V_{(1)}L) ⊗ S_{m} = V ⊗ S_{m}, since L, J^{s}, G^{i}, \overline{G}^{i} are primary eigenvectors of L with conformal weight greater than 1 for s = 1, 2, 3 and i = 1, 2. That is,

a_{(0)}L = (Δ − 1)∂_{A}a
a_{(1)}L = Δa
a_{(m)}L = 0

for all m > 1, a = L, J^{s}, G^{i}, \overline{G}^{i}, and some Δ = Δ(a) ≥ 1.

It is straightforward to verify that for ℓ ≥ 1 and s ∈ S_{m},

a ⊗ s_{(1)}∂_{A}^{(ℓ+1)}L ⊗ 1 = ((ℓ + 2)Δ − (ℓ + 1))∂_{A}^{(ℓ+1)}a ⊗ s + Δ∂_{A}^{(ℓ)}a ⊗ \frac{ds}{dt}.

Since Δ ≥ 1, we see that Δ∂_{A}^{(ℓ+1)}a ⊗ s ∈ ∑_{j=0}^{ℓ+1} V ⊗ S_{m}, ℓ ≥ 0}.

Step 2: Let B = L(A, σ) ⊆ Â for some finite order automorphism σ : A → A. Then B = Span_{k}{a_{(1)}∂_{A}^{(ℓ)}L ⊗ 1 | a ∈ B, ℓ ≥ 0}. Proof: Let Γ ⊆ Aut_{S_{m}}A_{m} be the cyclic subgroup of order m := |σ| generated by σ ⊗ ψ, where ψ is the R-automorphism of S_{m} given by sending t^{\frac{1}{m}} to ξ_{m}^{−1}t^{\frac{1}{m}} and ξ_{m} is the primitive mth root of 1 fixed in §0. Let

π : A_{m} → A_{m}
a ↦ \frac{1}{m} ∑_{i=0}^{m−1}(σ ⊗ ψ)^{i}(a).

Then B = A_{m}^{Γ}, the set of Γ-fixed points in A_{m}, and π is a surjection from A_{m} to B.

Since A_{m} = Span_{k}{v_{(1)}∂_{A}^{(ℓ)}L ⊗ 1 | v ∈ V ⊗ S_{m}, ℓ ≥ 0} and σ ⊗ ψ ∈ Aut_{S_{m}}(A_{m}) by Lemma 2.11, we have

B = π(A_{m}) = Span_{k} {π (v_{(1)}∂_{A}^{(ℓ)}L ⊗ 1) | v ∈ V ⊗ S_{m}, ℓ ≥ 0}

= Span_{k} {∑_{i=0}^{m−1} π(v)_{(1)}∂_{A}^{(ℓ)}σ^{i}(L) ⊗ 1 | v ∈ V ⊗ S_{m}, ℓ ≥ 0} .

By 3.66, there is a Y ∈ SL_{2}(S_{m}) such that

σ(G^{1}) ⊗ 1 = (σ ⊗ 1)(G^{1} ⊗ 1)

= c(Y^{−1})^{T} ( \begin{matrix}a \\ b \end{matrix} ) ⊗ ( \begin{matrix}1 \\ 0 \end{matrix} ) + dY ( \begin{matrix}0 & 1 \\ −1 & 0 \end{matrix} ) ( \begin{matrix}1 \\ 0 \end{matrix} ) ⊗ ( \begin{matrix}0 \\ 1 \end{matrix} ) ,

so Y ∈ SL_{2}(k). Then Y′ = 0, so σ(L) ⊗ 1 = (σ ⊗ 1)(L ⊗ 1) = L ⊗ 1 by (3.64). Hence σ(L) = L.

Therefore,

B = Span_{k} {∑_{i=0}^{m−1} π(v)_{(1)}∂_{A}^{(ℓ)}σ^{i}(L) ⊗ 1 | v ∈ V ⊗ S_{m}, ℓ ≥ 0}
= Span_{k} {π(v)_{(1)}∂_{A}^{(ℓ)}L ⊗ 1 | v ∈ V ⊗ S_{m}, ℓ ≥ 0}
⊆ Span_{k}{a_{(1)}∂_{A}^{(ℓ)}L ⊗ 1 | a ∈ B, ℓ ≥ 0}.

Step 3: Let χ ∈ Ctd_{k}(B), where B is as above. Then χ(L ⊗ 1) = L ⊗ r for some r ∈ R. Proof: By the argument in Step 2, every automorphism σ ∈ Aut_{k−conf}A fixes L, so L ⊗ 1 ∈ B. Then

L ⊗ 1_{(1)}χ(L ⊗ 1) = χ(L ⊗ 1_{(1)}L ⊗ 1) = 2χ(L ⊗ 1).

Taking an eigenspace decomposition of Â with respect to the operator

L ⊗ 1_{(1)} : a ⊗ s ↦ L ⊗ 1_{(1)}a ⊗ s = (L_{(1)}a) ⊗ s,

we have

Â = (⨁_{k=1}^{∞} Â_{k}) ⊕ (⨁_{ℓ=1}^{∞} Â_{\frac{1}{2} +ℓ}) ,  (3.72)

where

Â_{k} = Span {∂_{A}^{(k−2)}L ⊗ r, ∂_{A}^{(k−1)}J ⊗ r | J ∈ {J^{1}, J^{2}, J^{3}} and r ∈ Ŝ}
Â_{\frac{1}{2} +ℓ} = Span \{∂_{A}^{(ℓ−1)}G ⊗ r | G ∈ \{G^{1}, G^{2}, \overline{G}^{1}, \overline{G}^{2}\} and r ∈ Ŝ\}

are the eigenspaces with eigenvalues k and \frac{1}{2} + ℓ, respectively.

Thus χ(L ⊗ 1) ∈ Â_{2}, so χ(L ⊗ 1) = L ⊗ r + ∑_{i=1}^{3} ∂_{A}J^{i} ⊗ r_{i} for some r, r_{i} ∈ Ŝ. But also

0 = χ(L ⊗ 1_{(2)}L ⊗ 1)

= L ⊗ 1_{(2)}χ(L ⊗ 1)

= L ⊗ 1_{(2)} (L ⊗ r + ∑_{i=1}^{3} ∂_{A}J^{i} ⊗ r_{i})

= 2 ∑_{i=1}^{3} J^{i} ⊗ r_{i}.

Hence r_{i} = 0 for i = 1, 2, 3, so χ(L ⊗ 1) = L ⊗ r. Since χ ∈ Ctd_{k}(B) and σ(L) = L, we see that r ∈ R.

Step 4: Let χ and r be as in Step 3. Then χ (∂_{A}^{(k)}L ⊗ 1) = ∂_{A}^{(k)}L ⊗ r for all k ≥ 0. Proof: If b ∈ B is a primary eigenvector^{12} of itself with conformal weight Δ, then it is straightforward to verify that b_{(k+1)}∂_{B}^{(k)}b = (Δ + k − 1)∂_{B}^{(k)}b. By (CS4), we have

(Δ + k − 1)χ (∂_{B}^{(k)}b) = χ (b_{(k+1)}∂_{B}^{(k)}b)
= (−1)^{k}χ (∂_{B}^{(k)}b_{(k+1)}b)
= (−1)^{k}∂_{B}^{(k)}b_{(k+1)}χ(b).

If χ(b) = rb, this gives

(Δ + k − 1)χ (∂_{B}^{(k)}b) = r_{B}(−1)^{k}∂_{B}^{(k)}b_{(k+1)}b
= r_{B}b_{(k+1)}∂_{B}^{(k)}b
= (Δ + k − 1)r∂_{B}^{(k)}b,

12See definition in proof of Step 1.

so χ (∂_{B}^{(k)}b) = r_{B} ∘ ∂_{B}^{(k)}b if Δ > 1. Applying this result to b = L ⊗ 1, we see that

χ (∂_{A}^{(k)}L ⊗ 1) = χ (∂_{B}^{(k)}(L ⊗ 1))
= r_{B} ∘ ∂_{B}^{(k)}(L ⊗ 1)
= ∂_{A}^{(k)}L ⊗ r.

Step 5: In the notation of Step 3, χ = r_{B}. That is, Ctd_{k}(B) ⊆ R_{B}. Proof: For any a ∈ B, Step 4 shows that

χ (a_{(1)}∂_{A}^{(ℓ)}L ⊗ 1) = a_{(1)}χ (∂_{A}^{(ℓ)}L ⊗ 1)
= a_{(1)}∂_{A}^{(ℓ)}L ⊗ r
= r_{B} (a_{(1)}∂_{A}^{(ℓ)}L ⊗ 1) .

Since B is spanned by elements of the form a_{(1)}∂_{A}^{(ℓ)}L ⊗ 1 (Step 2), we see that χ = r_{B}.

Step 6: Two twisted loop algebras of A are R-isomorphic if and only if they are k-isomorphic. That is, (i) and (ii) are equivalent. Proof: Clearly R_{B} ⊆ Ctd_{R}(B) ⊆ Ctd_{k}(B). By Step 5, these inclusions are equalities. Thus (i) and (ii) are equivalent by Corollary 2.36.

From the eigenspace decomposition of Â, we see that the left multiplication operator L ⊗ 1_{(1)} : Â → Â is injective, and it follows easily that Z(B) = 0 for any twisted loop algebra B of A. Thus by Corollary 2.39, (ii) is equivalent to (iv).

It remains to check that (i), (iii), and (iv) are equivalent. By Theorem 2.16, Proposition 2.29, and Proposition 3.69, the R-isomorphism classes of Ŝ/R-forms of the R-conformal superalgebra A ⊗ R are parametrized by H_{ct}^{1}(ℤ̂, SL_{2}(Ŝ) × SL_{2}(k))/ ± (I, I)).

To simplify the notation, we write G = SL_{2}(Ŝ)×SL_{2}(k), \overline{G} = (SL_{2}(Ŝ)× SL_{2}(k))/ ± (I, I), and consider the exact sequence of (continuous) ℤ̂-groups

1 → ℤ/2ℤ → G → \overline{G} → 1.

where ℤ/2ℤ is identified with the subgroup of G generated by (−I, −I). This yields the exact sequence of pointed sets

H_{ct}^{1}(ℤ̂, ℤ/2ℤ) →^{α} H_{ct}^{1}(ℤ̂, G) →^{β} H_{ct}^{1}(ℤ̂, \overline{G}) →^{∂} H_{ct}^{2}(ℤ̂, ℤ/2ℤ).  (3.73)

whose individual terms can be understood as follows:

(a) H_{ct}^{1}(ℤ̂, ℤ/2ℤ) = Hom_{ct}(ℤ̂, ℤ/2ℤ) ≃ ℤ/2ℤ.

A representative cocycle of the nonzero element of this H^{1} is the (unique and continuous) map ℤ̂ → ℤ/2ℤ such that 1 ↦ (−I, −I).

(b) H_{ct}^{2}(ℤ̂, ℤ/2ℤ) = 0. This could be checked by brute force in terms of cocycles. An abstract argument is as follows. Since k is of characteristic 0 we have ℤ/2ℤ ≃ μ_{2}. Now H_{ct}^{2}(ℤ̂, μ_{2}) is part (in fact all of) the 2-torsion of the (algebraic) Brauer group H_{ét}^{2}(R, G_{m}). Because R is of cohomological dimension 1, this Brauer group vanishes.

(c) H_{ct}^{1}(ℤ̂, SL_{2}(Ŝ)) = 1. Indeed this H_{ct}^{1} is the part of H_{ét}^{1}(R, SL_{2}) corresponding to the isomorphism classes of R–torsors under SL_{2} that become trivial over Ŝ.^{13} As observed in Remark 2.30 H_{ét}^{1}(R, SL_{2}) vanishes.^{14}

From (c), we immediately obtain

(d) The canonical map H_{ct}^{1}(ℤ̂, SL_{2}(k)) → H_{ct}^{1}(ℤ̂, G) is bijective.

Finally, since ℤ̂ acts trivially on SL_{2}(k) we have

(e) H_{ct}^{1}(ℤ̂, SL_{2}(k)) ≃ {conjugacy classes of elements of finite order in SL_{2}(k)}.

13In fact all of H_{ét}^{1}(R, SL_{2}): Every R-torsor under SL_{2} is isotrivial. This follows from [16] in the case of R. More general isotriviality results can be found in [5].

14One can avoid the general considerations of [16] in the present case by the following direct argument. The exact sequence of R-groups 1 → SL_{2} → GL_{2} →^{det} G_{m} → 1 yields

GL_{2}(R) →^{det} R^{×} → H_{ét}^{1}(R, SL_{2}) → H_{ét}^{1}(R, GL_{2}).

Since the map det is surjective, the map H_{ét}^{1}(R, SL_{2}) → H_{ét}^{1}(R, GL_{2}) has trivial kernel.

On the other hand H_{ét}^{1}(R, GL_{2}) ≃ H_{Zar}^{1}(R, GL_{2}) = 1 (the first equality by Grothendieck-Hilbert 90, and the last since all rank 2 projective modules over R are free; because R is a principal ideal domain).

By (b) and (e), we have a surjective map

β : { conjugacy classes of elements of finite order in SL_{2}(k)}
⟶ H_{ct}^{1}(ℤ̂, \overline{G}).

Tracing through the various definitions, we see that the explicit nature of the map β is as follows: let θ ∈ SL_{2}(k) be of finite order. Define a cocycle u_{θ} ∈ Z^{1}(ℤ̂, \overline{G}) by u_{θ}(n) = \overline{(1, θ^{n})} where ^{−} : G → \overline{G} is the canonical map. Then β maps the conjugacy class of θ to the class of u_{θ} in H_{ct}^{1}(ℤ̂, \overline{G}).

It remains to show that u_{θ} and u_{σ} are cohomologous if and only if θ is conjugate to ±σ. If [u_{θ}] = [u_{σ}] there exists (x, τ) ∈ G = SL_{2}(Ŝ) × SL_{2}(k) such that

\overline{(x, τ)}^{−1}u_{θ}(n) ^{n} \overline{(x, τ)} = u_{σ}(n)

for all n ∈ ℤ ⊆ ℤ̂. In particular, for n = 1 we get

\overline{(x^{−1} ^{1}x, τ^{−1}θτ)} = \overline{(1, σ)}

which forces either

x^{−1} ^{1}x = 1  and τ^{−1}θτ = σ
or x^{−1} ^{1}x = −1 and τ^{−1}θτ = −σ.

Thus θ is conjugate to either σ or −σ. The converse is obvious given that

the element (\begin{matrix}t^{1/2} & 0 \\ 0 & t^{1/2} \end{matrix}) ∈ SL_{2}(Ŝ^{×}) satisfies x^{−1} ^{1}x = −1.

We have thus shown that H_{ct}^{1}(ℤ̂, \overline{G}) is in bijective correspondence with the conjugacy classes of elements of finite order in PGL_{2}(k). This completes the proof of the theorem. □

The grading operator L is stable under all automorphisms of the N = 2 and N = 4 conformal superalgebras, so L ⊗ 1 is an element of every twisted loop algebra of these conformal superalgebras. By considering the n-products of elements with L ⊗ 1, it is easy to verify that every twisted loop algebra of the N = 2 and N = 4 conformal superalgebras is centreless. They each admit a (unique) one-dimensional universal central extension, as was previously shown by one of the authors [11]. Using Corollary 2.39, we see that Theorems 3.35 and 3.71 actually give a parametrization of the kisomorphism classes of universal central extensions of twisted loop algebras of the N = 2 and N = 4 conformal superalgebras. Summarizing:

Corollary 3.74 There are exactly two ℂ-isomorphism classes of twisted loop algebras based on the N = 2 conformal superalgebra, and infinitely many ℂ-isomorphism classes of twisted loop algebras based on the N = 4 conformal superalgebra. The explicit automorphisms giving the distinct isomorphism classes are the identity map and the automorphism ω in the N = 2 case; and representatives of the conjugacy classes of elements of finite order in PGL_{2}(ℂ) in the N = 4 case.

Furthermore, two of these twisted loop algebras are isomorphic if and only if their universal central extensions are isomorphic. □

Remark 3.75 As explained in the introduction, the superconformal algebras in Schwimmer and Seiberg’s original work are obtained as formal distribution algebras of the twisted loop algebras of Lie conformal superalgebras. Since isomorphic twisted loop algebras lead to isomorphic superconformal algebras, our work shows that in the N = 2, 4 case there can be no more superconformal algebras than those listed by Schwimmer and Seiberg.

Remark 3.76 Our methods work equally well for the other N-conformal superalgebras: The isomorphism classes of loop algebras based on an Nconformal superalgebra A are parametrized by

H^{1}(Ŝ/R, Aut(A)) ≃ H_{ct}^{1}(π_{1}(R), Aut (A)(Ŝ)) ≃ H_{ct}^{1}(ℤ̂, Aut (A)(Ŝ))

For N = 0, the conformal superalgebra is Vir, and the group Aut (A)(Ŝ) is trivial. Thus the only twisted loop algebra is the affinization of A. For N = 1, we have Aut (A)(Ŝ) ≃ ℤ/2ℤ. There are thus two non-isomorphic twisted loop algebras (Ramond and Neveu-Schwarz). For N = 3, the group Aut (A)(Ŝ) would appear to be O_{3}(Ŝ).^{15} Under this assumption, the cohomology will yield two non-isomorphic twisted loop algebras–again Ramond and Neveu-Schwarz.

References

[1] B. Allison, S. Berman, A. Pianzola, Covering algebras II: Isomorphism of loop algebras, J. Reine Angew. Math. 571 (2004), 39–71.

[2] R. Borcherds, Vertex algebras, Kac-Moody algebras, and the Monster, Proc. Natl. Acad. Sci. USA 83 (1986), 3068–3071.

15The calculation is delicate, just as in the N = 4 case. We have not checked the details thoroughly.

[3] P. Gille and A. Pianzola, Isotriviality of torsors over Laurent polynomial rings, C. R. Acad. Sci. Paris, Ser. I 341 (2005), 725–729.

[4] P. Gille and A. Pianzola, Galois cohomology and forms of algebras over Laurent polynomial rings, Math. Annalen 338 (2007) 497–543.

[5] P. Gille and A. Pianzola, Isotriviality and étale cohomology of Laurent polynomial rings, J. Pure Appl. Algebra 212 (2008) 780–800.

[6] D. Grantcharov and A. Pianzola, Automorphisms and twisted loop algebras of finite-dimensional simple Lie superalgebras, Int. Math. Res. Not. 73 (2004), 3937–3962.

[7] D. Grantcharov and A. Pianzola, Automorphisms of toroidal Lie superalgebras, Jour. of Algebra 319 (2008) 4230-4248.

[8] V. Kac, Infinite Dimensional Lie Algebras, third edition, Cambridge University Press, Cambridge, 1990.

[9] V.G. Kac, Vertex Algebras for Beginners, Univ. Lecture Ser., vol. 10, Amer. Math. Soc., 1996. Second edition, 1998.

[10] V.G. Kac, Formal distribution algebras and conformal algebras, XIIth International Congress of Mathematical Physics (Brisbane, 1997), Int. Press, Cambridge, MA, 1999, pp. 80–97.

[11] V.G. Kac and J. van de Leur, On classification of superconformal algebras, Strings 88, World Scientific, Singapore, 1989, pp. 77–106.

[12] M.-A. Knus et M. Ojanguren Théorie de la Descente et Algèbres d’Azumaya, Lecture Notes in Math., vol. 389, Springer, New York, 1974.

[13] Z. Kobayashi, Automorphisms of finite order of the affine Lie algebra A_{l}^{(1)}, Tsukuba J. Math. 10 (1986), 269–283.

[14] B. Margaux, Passage to the limit in non-abelian Čech cohomology, Journal of Lie Theory 17 (2007) 591–596.

[15] D.H. Peterson and V. Kac, Infinite flag varieties and conjugacy theorems, Proc. Natl. Acad. Sci. USA 80 (1983), 1778–1782.

[16] A. Pianzola, Vanishing of H^{1} for Dedekind rings and applications to loop algebras, C. R. Acad. Sci. Paris, Ser. I 340 (2005), 633-638.

[17] A. Schwimmer and N. Seiberg, Comments on the N = 2, 3, 4 superconformal algebras in two dimensions, Phys. Lett. B 184 (1987), 191–196.

[18] Séminaire de Géométrie Algébrique de l’I.H.E.S., Revêtements étales et groupe fondamental, dirigé par A. Grothendieck, Lecture Notes in Math., vol. 224, Springer, New York, 1971.

[19] Séminaire de Géométrie Algébrique de l’I.H.E.S., 1963-1964, Schémas en Groupes, dirigé par M. Demazure et A. Grothendieck, Lecture Notes in Math., vol. 151–153, Springer, New York, 1970.

[20] W.C. Waterhouse, Introduction to Affine Group Schemes, Graduate Texts in Math., vol. 66, Springer, New York, 1979.