On Universal Central Extensions of 𝔰𝔩_{n}(A)

Michael Lau

Abstract. We describe the interplay between several (co)homology theories in the context of computing universal central extensions of Lie algebras of matrices over some associative algebras. We illustrate these methods with an explicit computation of the universal central extension of the Lie algebra of n × n matrices over a family of localizations of the Weyl algebra A_{r}.

1. Introduction

Central extensions arise in several contexts of mathematics and theoretical physics. They are used to enrich and elucidate representation theory, and they capture some anomalies that occur when classical theories are quantized. The universal central extensions satisfy a universal mapping property and play a prominent role in the representation theory of infinite-dimensional Lie algebras.

In most cases, however, computation of these extensions is a nontrivial problem in homological algebra. Simple and elegant presentations are known in great generality (cf. [BM], [KL], or [vdK], for instance), but it is seldom straightforward to explicitly compute even the dimension of a universal central extension in a concrete setting.

Much work has been done for central extensions of Lie algebras of matrices over commutative rings. (See [BK], [Bl], [Ga], [He], [Ka], or [Zu], for example.) These include the (untwisted) affine Lie algebras, the toroidal Lie algebras, and many algebras graded by abelian groups, including the algebras of currents studied by physicists.

In this short paper, we use the interplay between several (co)homology theories to calculate the universal central extensions of matrix Lie algebras over some associative algebras that are far from commutative. These algebras include the rings of differential operators on some nonsingular affine varieties. We illustrate the techniques with an explicit computation of the dimensions of the universal central extensions of the Lie algebras of matrices over a family of localizations of the Weyl algebra A_{r}.

1991 Mathematics Subject Classification. 19C09, 16E40, 16S32.

Key words and phrases. central extensions, cyclic homology, differential operators.

The author thanks Professors G. Benkart and E. Neher for their helpful suggestions.

Support from NSERC Discovery Grant #8836 and NSA grant MDA904-03-1-0068

is also gratefully acknowledged.

Throughout this paper, 𝕂 will be a field of characteristic zero, all tensor products will be over 𝕂, and all associative algebras will be assumed to be unital. The integers and nonnegative integers will be denoted ℤ and ℕ, respectively.

2. Central extensions and cyclic homology

2.1. Central extensions. In this section, we give a brief review of the basic definitions and theory of Lie algebra central extensions. More details can be found in [Ga], [MP], [Ne], or [vdK].

Let 𝔤 be an arbitrary Lie algebra over 𝕂. A central extension of 𝔤 is a Lie algebra 𝔤̃ and an epimorphism ϕ : 𝔤̃ → 𝔤 with kernel ker ϕ contained in the centre of 𝔤̃. Such a central extension is said to have kernel ker ϕ and dimension dim(ker ϕ). Given two central extensions (𝔤̂, ψ) and (𝔤̃, ϕ) of 𝔤, a morphism (from (𝔤̂, ψ) to (𝔤̃, ϕ)) in the category of central extensions is a Lie algebra homomorphism μ : 𝔤̂ → 𝔤̃ such that ϕ ∘ μ = ψ. We say that (𝔤̂, ψ) and (𝔤̃, ϕ) are isomorphic if the morphism μ is a bijection. The central extension (𝔤̂, π) is universal if there is a unique morphism from it to every other central extension of 𝔤. There is a well-known isomorphism between ker π and the (Chevalley-Eilenberg) homology group H_{2}(𝔤; 𝕂) of 𝔤 with coefficients in the trivial module 𝕂. (See [We] §7.7, for instance.)

Isomorphism classes of one-dimensional central extensions are in bijective correspondence with cohomology classes in H^{2}(𝔤; 𝕂). In particular, each class [c] in H^{2}(𝔤; 𝕂) determines a central extension 𝔤̃ = 𝔤 ⊕ 𝕂c with the bracket [ · , ·] in 𝔤̃ given by [x, y] = [x, y] + c(x, y)c for x, y ∈ 𝔤, where c is central and c is a representative of the class [c]. Conversely, if c : 𝔤 × 𝔤 → 𝕂 is 𝕂-bilinear and 𝔤̃ = 𝔤 ⊕ 𝕂c is a Lie algebra under the bracket [ · , ·] defined above, then c is a representative of a cohomology class [c] in H^{2}(𝔤; 𝕂). Direct computation with the chain complex defining H^{2}(𝔤; 𝕂) shows that [c] ∈ H^{2}(𝔤; 𝕂) if and only if c satisfies the 2-cocycle conditions:

(C1) c(x, y) = −c(y, x) and

(C2) c([x, y], z) + c([y, z], x) + c([z, x], y) = 0.

If 𝔤 has a universal central extension (𝔤̂, π), then by universality, there is a bijection between isomorphism classes of one-dimensional central extensions of 𝔤 and 𝕂-linear maps from ker π to 𝕂. That is, H^{2}(𝔤; 𝕂) ≅ Hom_{𝕂}(H_{2}(𝔤; 𝕂), 𝕂), which is precisely the isomorphism given by the universal coefficient theorem.

A Lie algebra 𝔤 is perfect if it is equal to its commutator subalgebra [𝔤, 𝔤]. The following lemma appears in van der Kallen [vdK]:

Lemma 2.1.

(1) If 𝔤̂ and 𝔤̃ are universal central extensions of a Lie algebra 𝔤, then 𝔤̂ ≅ 𝔤̃ as central extensions.

(2) If 𝔤̂ is the universal central extension of a Lie algebra 𝔤, then 𝔤̂ is perfect.

(3) A Lie algebra 𝔤 has a universal central extension if and only if 𝔤 is perfect.

□

Many of the most important infinite-dimensional Lie algebras, including the Heisenberg Lie algebras, the (untwisted) affine Lie algebras, the toroidal Lie algebras, and the Virasoro Lie algebra can be viewed as nontrivial central extensions.

2.2. Cyclic homology. As we have already noted, the kernel ker π of the universal central extension (L̂, π) is isomorphic to the homology group H_{2}(L; 𝕂). In the case where L = 𝔰𝔩_{n}(𝕂) ⊗ A for some commutative associative algebra A over 𝕂, this homology group is isomorphic as a vector space to a module of differential forms which depends only on A.

For any commutative associative algebra A, let Ω_{A}^{1} be the left A-module generated by the set of symbols dA = {da | a ∈ A} modulo the relations d(λa + b) = λda + db and d(ab) = a(db) + b(da) for all a, b ∈ A and λ ∈ 𝕂. Assuming only that 𝕂 is a commutative ring of characteristic not equal to 2, S. Bloch [Bl] showed that H_{2}(𝔰𝔩_{n}(𝕂) ⊗ A; 𝕂) ≅ Ω_{A}^{1}/dA for n ≥ 5. In [Ka], this is extended to the case where n ≥ 2, using our additional hypothesis that 𝕂 has characteristic 0.

Proposition 2.2. ^{1}

Let A be a commutative associative algebra (over 𝕂). Then as vector spaces,

H_{2}(𝔰𝔩_{n}(𝕂) ⊗ A; 𝕂) ≅ Ω_{A}^{1}/dA for n ≥ 2.

□

If A is an arbitrary associative algebra (not necessarily commutative), then let 𝔰𝔩_{n}(A) be the Lie algebra of n × n matrices X over A whose trace tr(X) ∈ [A, A], where [A, A] is the 𝕂-span of the elements ab − ba for a, b ∈ A. The Lie algebra 𝔰𝔩_{n}(A) is perfect for n ≥ 2, and its universal central extension is described by cyclic (co)homology, a noncommutative analogue of de Rham cohomology introduced in the early 1980s by A. Connes [Co].

For any associative algebra A over 𝕂, the cyclic complex C_{∗}(A) consists of the spaces C_{n}(A) = A^{⊗(n+1)}/im(τ −1) of coinvariants of the signed permutation action

(2.1)  τ = τ_{n} : a_{0} ⊗ ⋯ ⊗ a_{n} ↦ (−1)^{n}a_{n} ⊗ a_{0} ⊗ a_{1} ⊗ ⋯ ⊗ a_{n−1}

on the Hochschild complex A^{⊗(n+1)}, together with the usual Hochschild boundary maps

d : a_{0} ⊗⋯⊗a_{n} ↦ (−1)^{n}a_{n}a_{0} ⊗a_{1} ⊗⋯⊗a_{n−1} +∑_{i=0}^{n−1}(−1)^{i}a_{0} ⊗⋯⊗a_{i}a_{i+1} ⊗⋯⊗a_{n}.

Dualizing gives the cyclic cocomplex C^{∗}(A) = Hom_{𝕂}(C_{∗}(A), 𝕂) with coboundary maps ∂ : C^{n}(A) → C^{n+1}(A) defined by

(∂γ)((a_{0} ⊗ ⋯ ⊗ a_{n+1}) + im(τ − 1)) = γ(d((a_{0} ⊗ ⋯ ⊗ a_{n+1}) + im(τ − 1)))

for each γ ∈ Hom_{𝕂}(C_{n}(A), 𝕂). Its cohomology is called (Connes) cyclic cohomology, and is denoted HC^{∗}(A).

Motivated by the work of S. Bloch [Bl] in the commutative context, C. Kassel and J.-L. Loday [KL] substituted the cyclic homology group HC_{1}(A) for the module Ω_{A}^{1}/dA that appeared in Bloch’s commutative setting. Using a Morita equivalence argument, they proved the following theorem ([KL], Thm 1.7) for any associative algebra A over 𝕂. Their result also holds over fields of characteristic p ≠ 2.

Theorem 2.3. If n ≥ 3, then H_{2}(𝔰𝔩_{n}(A); 𝕂) ≅ HC_{1}(A). □

1This result was also generalized by Kassel [Ka] to include Lie algebras of the form 𝔤 ⊗ A for any classical Lie algebra 𝔤 and commutative associative algebra A. Berman and Krylyuk [BK] later extended Kassel’s work to the twisted setting.

It is interesting to note that this result does not hold for n = 2. A counterexample appears in [Gao].

For n ≥ 2, the Lie algebra 𝔰𝔩_{n}(A) is perfect, but it need not be centreless. In general, the centre of 𝔰𝔩_{n}(A) is the set of central scalar matrices, {aI | a ∈ Z(A) ∩ [A, A]}, where Z(A) is the centre of A. In fact, 𝔰𝔩_{n}(A) is a (dimension dim(Z(A) ∩ [A, A])) central extension of the centreless quotient algebra 𝔰𝔩_{n}(A)/Z(𝔰𝔩_{n}(A)). This quotient is isomorphic to (𝔰𝔩_{n}(𝕂) ⊗ A) ⊕ ad [A, A], which has multiplication

(2.2) [x ⊗ a, y ⊗ b] = [x, y] ⊗ \frac{1}{2}(a ∘ b) + (x ∘ y) ⊗ \frac{1}{2}[a, b] + \frac{(x | y)}{n}ad [a, b]
(2.3) [ad a, x ⊗ b] = x ⊗ [a, b] = −[x ⊗ b, ad a]
(2.4) [ad a, ad b] = ad [a, b]

where x, y ∈ 𝔰𝔩_{n}(𝕂), a, b ∈ A, [a, b] = ab − ba, a ∘ b = ab + ba, (x | y) = tr(xy),

[x, y] = xy − yx and x ∘ y = xy + yx − \frac{2}{n}tr(xy)I_{n}.

More precisely, any element x in 𝔰𝔩_{n}(A) may be written as the sum of the

matrix x_{0} = x − \frac{1}{n}tr(x)I of trace 0 and the matrix \frac{1}{n}tr(x)I where tr(x) ∈ [A, A]. Then the map ψ : 𝔰𝔩_{n}(A) → (𝔰𝔩_{n}(𝕂) ⊗ A) ⊕ ad [A, A] given by

(2.5)  ψ(x) = x_{0} + ad tr(x)

is a well-defined Lie algebra homomorphism. It is clearly a surjection and ker ψ = {aI | a ∈ [A, A] ∩ Z(A)} = Z(𝔰𝔩_{n}(A)). The universal central extension of (𝔰𝔩_{n}(𝕂) ⊗ A) ⊕ ad [A, A] is the same as the universal central extension of 𝔰𝔩_{n}(A), and thus its kernel is isomorphic to the space (Z(A) ∩ [A, A]) ⊕ HC_{1}(A). Computing directly at the chain level, we see that

HC_{1}(A) = {∑_{i} a_{i} ⊗ b_{i} + S_{A} ∈ {A, A} | ∑_{i}[a_{i}, b_{i}] = 0}

where S_{A} = Span{a ⊗ b + b ⊗ a, ab ⊗ c + bc ⊗ a + ca ⊗ b | a, b, c ∈ A} and {A, A} = (A ⊗ A)/S_{A}. Then (Z(A) ∩ [A, A]) ⊕ HC_{1}(A) ≅ HF (A) where

HF (A) = {∑_{i} a_{i} ⊗ b_{i} + S_{A} ∈ {A, A} | ∑_{i} ad [a_{i}, b_{i}] = 0} .

Theorem 2.4 (Allison, Benkart, and Gao [ABG]). Let A be an arbitrary associative algebra, and assume n ≥ 3. Let L be the centreless Lie algebra (𝔰𝔩_{n}(𝕂) ⊗ A) ⊕ ad [A, A] with Lie bracket given by (2.2), (2.3), and (2.4). Then the universal central extension of L is (L̂, π̂) where L̂ = (𝔰𝔩_{n}(𝕂) ⊗ A) ⊕ {A, A} and L̂ has Lie bracket

[x ⊗ a, y ⊗ b] = [x, y] ⊗ \frac{1}{2}(a ∘ b) + (x ∘ y) ⊗ \frac{1}{2}[a, b] + tr(xy)\{a, b\}

[{a, b}, x ⊗ c] = x ⊗ π̂({a, b})c
[{a, b}, {c, d}] = {π̂({a, b})c, d} + {c, π̂({a, b})d},

where x, y ∈ 𝔰𝔩_{n}(𝕂), a, b, c, d ∈ A, {a, b} = a⊗b+S_{A} ∈ (A ⊗ A)/S_{A} = {A, A}. The

covering map π̂ : L̂ → L is given by π̂(x ⊗ a) = x ⊗ a and π̂(\{a, b\}) = \frac{1}{n}ad [a, b], and the kernel of π̂ is HF (A).

3. Connes sequence and Künneth theorem

For any (unital) associative algebra A, the cyclic cochains are the invariants of the natural action (2.1) of the signed cyclic permutations τ = τ_{n} on the spaces Hom_{𝕂}(A^{⊗(n+1)}, 𝕂). There is a vector space isomorphism

ψ : Hom_{𝕂}(A^{⊗(n+1)}, 𝕂) → Hom_{𝕂}(A^{⊗n}, A^{∗})

given by

(ψf)(a_{1} ⊗ ⋯ a_{n})(a) = f(a ⊗ a_{1} ⊗ ⋯ ⊗ a_{n})

for all f ∈ Hom_{𝕂}(A^{⊗(n+1)}, 𝕂) and a, a_{1}, … a_{n} ∈ A.

The space A^{∗} = Hom_{𝕂}(A, 𝕂) has an A-bimodule structure given by

(3.1)  (aγb)(c) = γ(bca) for γ ∈ A^{∗} and a, b, c ∈ A.

This lets us interpret Hom_{𝕂}(A^{⊗n}, A^{∗}) as a Hochschild cocomplex C_{H}^{∗}(A; A^{∗}) with coboundary map b : C_{H}^{n}(A; A^{∗}) → C_{H}^{n+1}(A; A^{∗}) given by

(bγ)(a_{1} ⊗ ⋯ ⊗ a_{n+1})

= a_{1}γ(a_{2} ⊗ ⋯ ⊗ a_{n+1}) + ∑_{i=1}^{n}(−1)^{i}γ(a_{1} ⊗ ⋯ ⊗ a_{i}a_{i+1} ⊗ ⋯ ⊗ a_{n+1})
+ (−1)^{n+1}γ(a_{1} ⊗ ⋯ ⊗ a_{n})a_{n+1}

for γ ∈ C_{H}^{n}(A; M).

Note that some care is needed to properly interpret the terms in the low dimensions. The space C_{H}^{0}(A; A^{∗}) = Hom_{𝕂}(A, A^{∗}) can be identified with A^{∗} by identifying each γ ∈ C_{H}^{0}(A; A^{∗}) with γ(1) ∈ A^{∗}. Then the proper interpretation of the coboundary b : C_{H}^{0}(A; A^{∗}) → C_{H}^{1}(A; A^{∗}) is (bγ)(a) = aγ − γa for any a ∈ A and γ ∈ A^{∗}.

Since the cyclic coboundary ψ ∘ ∂ ∘ ψ^{−1} is the restriction of the Hochschild coboundary b to the image ψ(C^{∗}(A)) of the cyclic cocomplex in C_{H}^{∗}(A; A^{∗}), ψ gives an embedding of C^{∗}(A) into C_{H}^{∗}(A; A^{∗}) as chain cocomplexes. The embedding gives a short exact sequence

0 ⟶ C^{∗}(A) ⟶^{ι} C_{H}^{∗}(A; A^{∗}) ⟶ \frac{C_{H}^{∗}(A; A^{∗})}{ι(C^{∗}(A))} ⟶ 0.

Connes ([Co], Cor 38) observed that the nth cohomology group of the quotient

complex \frac{C_{H}^{∗}(A; A^{∗})}{ι(C^{∗}(A))} is isomorphic to HC^{n−1}(A), so the short exact sequence in-

duces the following long exact sequence, called the Connes sequence, relating the cyclic cohomology HC^{∗}(A) to the Hochschild cohomology H^{∗}(A; A^{∗}):

Theorem 3.1. Let A be an associative algebra over the field 𝕂. Then the following is a long exact sequence.

⋯ ⟶ H^{n}(A; A^{∗}) ⟶ HC^{n−1}(A) ⟶ HC^{n+1}(A) ⟶ H^{n+1}(A; A^{∗}) ⟶ ⋯

□

A trivial consequence of the Connes sequence is that HC^{0}(A) ≅ H^{0}(A; A^{∗}). By definition, the Hochschild cohomology group

H^{0}(A; A^{∗}) = \frac{ker (b : C_{H}^{0}(A, A^{∗}) → C_{H}^{1}(A, A^{∗}))}{im (b : 0 → C_{H}^{0}(A, A^{∗}))}
= ker (b : C_{H}^{0}(A, A^{∗}) → C_{H}^{1}(A, A^{∗}))
= {γ ∈ A^{∗} | aγ − γa = 0 for all a ∈ A}
= {γ ∈ A^{∗} | γ(ba) − γ(ab) = 0 for all a, b ∈ A},

and this expression is clearly 0 if A = [A, A] as a Lie algebra under the canonical bracket [a, b] = ab − ba. When A = [A, A], we say that A is Lie-perfect. In this case, HC^{0}(A) ≅ H^{0}(A; A^{∗}) = 0, so the Connes sequence gives HC^{1}(A) ≅ H^{1}(A; A^{∗}). Another straightforward computation at the chain level shows that

H^{1}(A; A^{∗}) = \frac{Der(A, A^{∗})}{Inn(A, A^{∗})}, where Der(A, A^{∗}) (resp. Inn(A, A^{∗})) is the space of

derivations^{2} d (resp. inner derivations) from A to the A-bimodule A^{∗}. Since we are working over a field, HC^{k}(A) ≅ Hom_{𝕂}(HC_{k}(A), 𝕂) (for every k) by the universal coefficient theorem. This proves the following useful proposition:

Proposition 3.2. Let A be an associative algebra that is Lie-perfect. Then as vector spaces,

HC_{0}(A) = HC^{0}(A) = 0 and HC^{1}(A) ≅ \frac{Der(A, A^{∗})}{Inn(A, A^{∗})}.

□

We will also use the Künneth theorem for cyclic homology (cf. [Lo], Cor 4.3.12):

Theorem 3.3. Let A and B be associative algebras (over 𝕂). Then there is a natural long exact sequence:

⋯ ⟶ HC_{n}(A⊗B) ⟶ ⨁_{r+s=n} HC_{r}(A) ⊗ HC_{s}(B) ⟶ ⨁_{i+j=n−2}HC_{i}(A)⊗HC_{j}(B)

⟶ HC_{n−1}(A ⊗ B) ⟶ ⨁_{r+s=n−1} HC_{r}(A) ⊗ HC_{s}(B) ⟶ ⋯

□

4. Localized Weyl algebras

For r ≥ 1 and 0 ≤ s ≤ r, let A_{r,s} be the (unital) associative algebra generated by {p_{i}, q_{i}, q_{j}^{−1} | 1 ≤ i ≤ r and 1 ≤ j ≤ s}, modulo the relations

[p_{k}, q_{ℓ}] = δ_{k,ℓ}, [p_{k}, p_{ℓ}] = [q_{k}, q_{ℓ}] = 0, and q_{m}q_{m}^{−1} = q_{m}^{−1}q_{m} = 1

for all 1 ≤ k, ℓ ≤ r and 1 ≤ m ≤ s. Then A_{r,s} is the Weyl algebra A_{r} = A_{r,0}, localized at the multiplicative set {q_{1}^{a_{1}}q_{2}^{a_{2}} ⋯ q_{s}^{a_{s}} | a_{1}, a_{2}, …, a_{s} ∈ ℕ}. An easy computation shows that A_{r,s} is central simple—that is, its centre Z(A_{r,s}) = 𝕂1, and it has no nontrivial ideals. It has a natural irreducible, faithful representation

on 𝕂[t_{1}^{±1}, …, t_{s}^{±1}, t_{s+1}, …, t_{r}] by letting the p_{i} act as differentiation operators \frac{∂}{∂t_{i}}

and the q_{i}^{±1} act as multiplication by t_{i}^{±1}. If 𝕂 is algebraically closed, then A_{r,s} is also the ring of differential operators on the variety \underbrace{S^{1} × ⋯ × S^{1}}_{s terms} × \underbrace{𝕂 × ⋯ × 𝕂}_{r−s terms},

2d : A → A^{∗} is a derivation if d(ab) = (da).b + a.(db) for each a, b ∈ A. The map d is inner if there is a fixed γ ∈ A^{∗} such that d(a) = γ.a − a.γ for all a ∈ A.

where S^{1} is the circle in 𝕂^{2} given by the equation x^{2} + y^{2} = 1 (cf. [MR], §15.5).

For x = q or p, we use the multi-index notation x^{α} to denote x_{1}^{α_{1}}x_{2}^{α_{2}} ⋯ x_{r}^{α_{r}} for α = (α_{1}, α_{2}, …, α_{r}) ∈ ℤ^{s} × ℕ^{r−s}. For each γ ∈ ℕ^{r}, we write γ! for the product

γ_{1}!γ_{2}! ⋯ γ_{r}!, and \binom{α}{γ} for the product ∏_{i=1}^{r} \binom{α_{i}}{γ_{i}}. It is straightforward to verify the

following lemma using induction:

Lemma 4.1. Let α ∈ ℕ^{r} and β ∈ ℤ^{s} × ℕ^{r−s}. Then in A_{r,s},

p^{α}q^{β} = ∑_{γ∈ℕ^{r}} γ!\binom{α}{γ}\binom{β}{γ}q^{β−γ}p^{α−γ}.

□

Lemma 4.1 lets us write any element of A_{r,s} as a 𝕂-linear combination of terms of the form q^{α}p^{β}. Each element has a unique such representation, called its standard form. For each standard form monomial q^{α}p^{β} of A_{r,s}, we have

[p_{i}, q^{α}p^{β}] = α_{i}q^{α−e_{i}}p^{β}

and

[q_{i}, q^{α}p^{β}] = −β_{i}q^{α}p^{β−e_{i}},

where e_{i} = (0, …, 1, …, 0) with the nonzero entry 1 in the ith position. Since each element in A_{r,s} has a unique standard form expression, we say that

(4.1)  ad p_{i} = \frac{∂}{∂q_{i}}
(4.2)  ad q_{i} = −\frac{∂}{∂p_{i}}

as operators on A_{r,s} (for 1 ≤ i ≤ r).

Since the standard form monomials

q^{α}p^{β} = [−\frac{1}{β_{1} + 1}q_{1}, q^{α}p^{β+e_{1}}]

span A_{r,s}, we see that

Lemma 4.2. A_{r,s} is Lie-perfect. □

Let W be the associative algebra A_{1,1}. Viewed as a Lie algebra in the usual way, W arises as a limit of objects called W_{n}-algebras (see [Za], for instance), and is sometimes denoted W_{1+∞}. Observe that

A_{r,s} = \underbrace{W ⊗ ⋯ ⊗ W}_{s terms} ⊗ \underbrace{A_{1} ⊗ ⋯ ⊗ A_{1}}_{r−s terms},

where A_{1} is the Weyl algebra A_{1,0}.

By Proposition 3.2 and Lemma 4.2, HC_{0}(A_{r,s}) = 0 for every r and s. We use this fact, the Künneth theorem, and the decomposition A_{r,s} = A_{r−1,s−1} ⊗ W to compute HC_{1}(A_{r,s}) for r ≥ 2, s ≥ 1:

Proposition 4.3. If r ≥ 2 and s ≥ 1, then HC_{1}(A_{r,s}) = 0.

Proof. Let A = A_{r−1,s−1}. Then by the Künneth theorem for cyclic homology (Theorem 3.3), the following is part of a long exact sequence:

⨁_{k+ℓ=0} HC_{k}(A) ⊗ HC_{ℓ}(W) ⟶ HC_{1}(A ⊗ W) ⟶ ⨁_{k+ℓ=1} HC_{k}(A) ⊗ HC_{ℓ}(W).

But since A and W are Lie-perfect, HC_{0}(A) = HC_{0}(W) = 0, so both sums in the sequence above are 0. Hence HC_{1}(A_{r,s}) = HC_{1}(A ⊗ W) = 0. □

It remains only to compute HC_{1}(A) for A = A_{r} = A_{r,0} and A = A_{1,1} = W, which we will do using a derivations computation and another (fairly trivial) application of the Künneth theorem.

For our derivations computations, we will need to define the formal integral, a linear operator on A_{1} (resp. W). For any k, ℓ ∈ ℕ (resp. k ∈ ℤ and ℓ ∈ ℕ), let

∫ q^{k}p^{ℓ} dp = \frac{1}{ℓ + 1}q^{k}p^{ℓ+1},
∫ q^{k}p^{ℓ} dq = \{ \begin{matrix}\frac{1}{k+1}q^{k+1}p^{ℓ} & if k ≠ −1 \\ 0 & otherwise. \end{matrix}

Proposition 4.4. dim HC_{1}(A_{1}) = 0 and dim HC_{1}(W) = 1.

Proof. By Lemma 4.2, Proposition 3.2, and the universal coefficient theorem, it is enough to show that every derivation from A_{1} to A_{1}^{∗} is inner, and that Der(W, W^{∗}) = Inn(W, W^{∗}) ⊕ 𝕂D for some non-inner derivation D. We begin by identifying such a non-inner derivation D ∈ Der(W, W^{∗}) \ Inn(W, W^{∗}).

We will write p, q, q^{−1} for the generators p_{1}, q_{1}, q_{1}^{−1}, respectively, of the algebras A_{1} or W. Observe that any derivation η ∈ Der(W, W^{∗}) must satisfy 0 = η(1) = η(qq^{−1}) = (ηq)q^{−1}+q(ηq^{−1}), so ηq^{−1} = −q^{−1}(ηq)q^{−1}. The relation q^{−1}q = 1 forces the same formula for ηq^{−1} in terms of ηq. Thus any W^{∗}-valued map defined on the generators p and q can be extended to a derivation in Der(W, W^{∗}) if its (unique) extension respects the relation pq − qp = 1.

Let D(p) = 0 and D(q)(q^{r}p^{s}) = δ_{r,−1}δ_{s,0} for all r ∈ ℤ and s ∈ ℕ. Then for any such r and s,

D(pq − qp)(q^{r}p^{s}) = ((Dp)q + p(Dq) − (Dq)p − q(Dp))(q^{r}p^{s})
= (Dq)([q^{r}p^{s}, p]) = −(Dq)(rq^{r−1}p^{s}) = 0.

(Here we have used the definition of W^{∗} as a W-bimodule (Equation (3.1)).) Thus D defines a derivation in Der(W, W^{∗}).

Suppose D is inner. Then there exists some λ ∈ W^{∗} such that Dq = λq − qλ. But 1 = (Dq)(q^{−1}) = (λq − qλ)(q^{−1}) = λ(qq^{−1} − q^{−1}q) = 0, a contradiction. Hence D ∈ Der(W, W^{∗}) \ Inn(W, W^{∗}).

For the remainder of the proof, let A = A_{1} or W be fixed. For the sake of notational convenience, we will view D as a derivation in Der(A, A^{∗}) by defining D = 0 if A = A_{1}. Let ∂ ∈ Der(A, A^{∗}), and define γ = ∂ − (∂q)(q^{−1})D. It suffices to show that γ ∈ Inn(A, A^{∗}).

Let m(x) = −γ(q)(∫ x dp) for each x ∈ A. Note that m ∈ A^{∗} and μ(x) = m ∘ ad x = mx − xm defines an inner derivation μ ∈ Inn(A, A^{∗}). We will show that γ = μ.

For any y ∈ A_{r}, we see that y = \frac{∂c}{∂p} where c = ∫ y dp. Then using (4.1) and (4.2), we have

μ(p)(y) = m([p, y])
= m (\frac{∂^{2}c}{∂q∂p})
= m (\frac{∂^{2}c}{∂p∂q} )
= −γ(q) (∫ \frac{∂^{2}c}{∂p∂q} dp)
= −γ(q) (\frac{∂c}{∂q} + f(q)) .

for some Laurent polynomial f ∈ 𝕂[q, q^{−1}].

Since γ is a derivation, 0 = γ(1) = γ(0). Thus for every a ∈ A,

0 = γ([p, q])(a) = (γ(p)q + pγ(q) − γ(q)p − qγ(p))(a)
= γ(p)[q, a] − γ(q)[p, a].

Then by (4.1) and (4.2),

(4.3)  γ(p) (\frac{∂a}{∂p}) = −γ(q) (\frac{∂a}{∂q}).

Applying (4.3) gives

μ(p)(y) = −γ(q) (\frac{∂c}{∂q} + f(q))
= −γ(q) (\frac{∂c}{∂q} ) − γ(q) (\frac{∂}{∂q} ∫ f(q) dq)
= γ(p) (\frac{∂c}{∂p}) + γ(p) (\frac{∂}{∂p} ∫ f(q) dq)
= γ(p) (\frac{∂c}{∂p})
= γ(p)(y).

By similar arguments,

μ(q)(y) = γ(q)(y).

Since μ and γ agree on the generators of A, the derivations γ and μ are equal. In particular, γ ∈ Inn(A, A^{∗}). □

Since A_{r} ≅ \underbrace{A_{1} ⊗ ⋯ ⊗ A_{1}}_{r terms}, the following corollary now follows from the Künneth

theorem (Theorem 3.3):

Corollary 4.5. dim HC_{1}(A_{r}) = 0. □

Combining Propositions 4.3 and 4.4, and Corollary 4.5, we obtain the following theorem.

Theorem 4.6. For any r ≥ 1 and 0 ≤ s ≤ r,

HC_{1}(A_{r,s}) ≅ \{ \begin{matrix}𝕂 & if (r, s) = (1, 1) \\ 0 & otherwise. \end{matrix}

□

Since A_{r,s} is central simple and Lie-perfect, Theorem 4.6 also gives the skewdihedral homology HF (A_{r,s}) ≅ Z(A_{r,s}) ∩ [A_{r,s}, A_{r,s}].

Corollary 4.7.

dim HF (A_{r,s}) = \{ \begin{matrix}2 & if (r, s) = (1, 1) \\ 1 & otherwise. \end{matrix}

□

Explicit formulas for the 2-cocycles corresponding to the universal central extension of the centreless Lie algebra (𝔰𝔩_{n}(𝕂) ⊗ A_{r,s}) ⊕ ad A_{r,s} may be found in an elementary (though tedious) way (cf. [La2]) by choosing a basis for the skewdihedral homology and then computing the projection formulas associated with the splitting {A, A} = ad [A, A] ⊕ HF (A), in the notation of Theorem 2.4.

As we noted in the discussion before Theorem 2.4, the Lie algebra 𝔰𝔩_{n}(A_{r,s}) is a central extension of (𝔰𝔩_{n}(𝕂) ⊗ A_{r,s}) ⊕ ad A_{r,s}, and this central extension is universal if (r, s) ≠ (1, 1) by Corollary 4.7. In Theorems 4.8 and 4.9 below, we give an explicit formula for the 2-cocycle corresponding to this central extension.

We also give a formula for the more interesting cocycle λ corresponding to the universal central extension of 𝔰𝔩_{n}(A_{1,1}). This cocycle was first described by Kac-Peterson [KP], though they do not indicate their method of computation. A representation-theoretic approach appears in [La1] and a conformal algebra approach in [Ro]. A description of the universal central extension using generating functions is given in [VV]. The universality of the resulting extensions follows from Corollary 4.7 and Theorem 2.4. Li [Li] was the first to give a proof of the universality of the extension in Theorem 4.9.

Theorem 4.8. Let L = (𝔰𝔩_{n}(𝕂) ⊗ A_{r,s}) ⊕ ad A_{r,s} where (r, s) ≠ (1, 1) and n ≥ 3. Then the map μ : L × L → 𝕂 given by

(i) μ(x ⊗ a, y ⊗ b) = (x|y)⟨a, b⟩

(ii) μ(ad a, ad b) = n⟨a, b⟩
(iii) μ(x ⊗ a, ad b) = μ(ad b, x ⊗ a) = 0

is a 2-cocycle defining the universal central extension of L, where (x|y) is the trace T r(xy) and

⟨q^{α}p^{β}, q^{γ}p^{η}⟩ = \{ \begin{matrix}(γ)_{β} δ_{α+γ,β} & if β ≠ 0 and η = 0 \\ −(α)_{η} δ_{α+γ,η} & if β = 0 and η ≠ 0 \\ 0 & otherwise, \end{matrix}

where α, γ ∈ ℤ^{s} × ℕ^{r−s}, β, η ∈ ℕ^{r}, and

(α)_{η} = ∏_{i=1}^{r}(α_{i})(α_{i} − 1) ⋯ (α_{i} − (η_{i} − 1)).

□

Theorem 4.9. Let n ≥ 3 and W = A_{1,1}. Then the universal central extension of L = (𝔰𝔩_{n}(𝕂) ⊗ W) ⊕ ad W is L̃ := L ⊕ 𝕂c ⊕ 𝕂d with bracket [ · , ·] given by

[α, β] = [α, β] + λ(α, β)c + μ(α, β)d
c, d ∈ Z(L̃)

where [· , ·] is the bracket of the Lie algebra L defined in the discussion before Theorem 2.4, and λ, μ : L × L → 𝕂 are the 2-cocycles defined by the following formulas (in the notation of Theorem 4.8):

(1a) λ(x ⊗ a, y ⊗ b) = (x | y)⟨a, b⟩_{λ}

(1b) λ(ad a, ad b) = n⟨a, b⟩_{λ}
(1c) λ(x ⊗ a, ad b) = λ(ad b, x ⊗ a) = 0

and

(2a) μ(x ⊗ a, y ⊗ b) = (x|y)⟨a, b⟩_{μ}
(2b) μ(ad a, ad b) = n⟨a, b⟩_{μ}
(2c) μ(x ⊗ a, ad b) = λ(ad b, x ⊗ a) = 0

where

⟨q^{k}p^{ℓ}, q^{r}p^{s}⟩_{λ} = (−1)^{s+1}ℓ!s!\binom{r}{ℓ + s + 1}δ_{k+r,ℓ+s}

⟨q^{k}p^{ℓ}, q^{r}p^{s}⟩_{μ} = \{ \begin{matrix}(r)_{ℓ}δ_{k+r,ℓ} & if ℓ ≠ 0 and s = 0 \\ −(k)_{s}δ_{k+r,s} & if ℓ = 0 and s ≠ 0 \\ 0 & otherwise. \end{matrix}

□

5. Conclusion

Since the ring of differential operators on any nonsingular affine variety is a homomorphic image of a subalgebra of a Weyl algebra A_{r}, the techniques described in this paper should be applicable to a large family of matrix Lie algebras. Our restriction to characteristic 0 is necessary, however, since rings of differential operators need not be Lie-perfect in positive characteristic. For example, if the base field 𝕂 has characteristic p > 0, then none of the algebras A_{r,s} satisfies the Lie-perfect condition. But in characteristic 0, we conjecture that the ring of differential operators on any nonsingular affine variety is Lie-perfect, and thus amenable to the approach described in this paper.

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Department of Mathematics and Statistics, University of Ottawa, 585 King Edward Ave., Ottawa, Ontario K1N 6N5, Canada

    E-mail address: mlau228@mathstat.uottawa.ca