The Central Extension Defining the Super Matrix Generalization of W 1 ∞

We prove that the Lie superalgebra of regular differential 
operators on the superspace has an essentially unique non-trivial 
central extension.


Introduction
The W infinity algebras naturally arise in various physical systems, such as two-dimensional quantum gravity and the quantum Hall effects see the review 1, 2 and references there in .The most fundamental one is the W 1 ∞ which is the central extension of the Lie algebra of regular differential operators on the circle 1-5 , and it contains the W ∞ algebra as a subalgebra.Various extensions where constructed: super extension

, and the most general super matrix generalization
presented in 1, 2, 9 .It seems difficult to decide where and when the first definition of a version of super-W algebra appeared, but a book by Guieu and Roger 10 has a good historical and bibliographic base, including the pioneering papers of Radul where the superanalogues of the Bott-Virasoro cocycles were introduced see 11 .The original W 1 ∞ corresponds to M 1, N 0. The general study of representation theory of W infinity algebras started in the remarkable work 4 by Kac and Radul and continued in several works some of them are 6, 12-14 .Matrix generalizations are deeply related to the main examples of infinite rank conformal algebras see 15-17 .The super matrix generalization W M|N 1 ∞ is defined as a specific central extension of the Lie superalgebra of regular differential operators on the superspace C M|N t, t −1 .Only in the special case of W 1 ∞ i.e., M 1, N 0 was it proved that the 2-cocycle defining this central extension is unique up to coboundary 18 .The main goal of the present work is to extend

Basic Definitions and Main Result
Let L and L be two Lie superalgebras over C. The Lie superalgebra L is said to be a onedimensional central extension of L if L is the direct sum of L and CC as vector spaces and the Lie superbracket in L is given by for all a, b ∈ L, where •, • is the Lie bracket in L and Ψ : L × L → C is a 2-cocycle on L, that is, a bilinear C-valued form satisfying the following conditions for all homogeneous elements a, b, c ∈ L: Now, we will introduce the Lie superalgebra that will be considered in this work.Let us denote by Mat M | N the associative superalgebra of linear transformations on the complex M | N -dimensional superspace C M|N .Namely, we consider the set of all M N × M N matrices of the form where matrices, respectively, with complex entries.The Z 2 -gradation is defined by declaring that matrices of the form 2.4 with A A − 0 are even, and those with A 0 A 1 0 are odd.We denote by |A| the degree of A with respect to this Z 2 -gradation.The supertrace is defined by and it satisfies Str AB −1 |A||B| Str BA .
Let D as be the associative algebra of regular differential operators on the circle, that is, the operators on C t, t −1 of the form 2.6 The elements form its basis, where ∂ t denotes d/dt.Another basis of D as is where D t∂ t .It is easy to see that Here and further we use the notation SD M|N ⊕ CC be the central extension of SD M|N by a one-dimensional vector space with a specified generator C, whose commutation relation for homogeneous elements is given by where the 2-cocycle Ψ is given by

2.12
Now, we are in condition to state our main result.

Proof of Theorem 2.1
We will need the explicit expression of the bracket of basis elements of type 2.9 in SD M|N :

3.1
In particular, we have

3.2
Let β be a 2-cocycle on SD M|N .We consider the linear functional in SD M|N defined by

3.3
Then β 1 β − α f β is a 2-cocycle on SD M|N that is equivalent to β, and using 3.3 , we obtain

3.4
In order to complete the proof we need to show that Ψ aβ 1 for some a ∈ C. By observing the supertrace that appears in the expression of Ψ in 2.12 , we immediately obtain that for any f, g ∈ D as Ψ fE ij , gE sk 0 if i / k or j / s.

3.5
In Lemmas 3.1 and 3.2, we will show that β 1 also satisfies 3.5 .Case j / i and s i.

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In this case we have • g E ij 0 using i / j and 3.6 .

3.7
Case j / i and s / i.
By taking the usual bracket, we make the associative algebra D as into a Lie algebra which is denoted by D. Observe that It is easy to show that D, D D; therefore, for any f ∈ D, we have Thus, if j / i and s / i, using 2.2 ,

3.10
The proof is finished.

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Lemma 3.2.For any f, g ∈ D as and i / j, s / k, β 1 fE ij , gE sk 0 when i / k or j / s.
Proof.If i / j and k / i, we have

3.11
Hence we have β 1 fE ij , gE sk 0. Finally, using skew-symmetry and the previous case, if i / j, s / k, and s / j, we have that β 1 fE ij , gE sk 0. Now, it remains to consider the expression β 1 fE ij , gE ji .In order to do it, consider again the Lie algebra D SD 1|0 see 3.8 and denote by ψ D the 2-cocycle Ψ defined in 2.12 with M 1 and N 0.
In fact, from the expression of Ψ, we have 3.12 Lemma 3.3.There exist a i ∈ C such that for all f, g ∈ D as β 1 fE ii , gE ii a i ψ D f, g .

3.13
Moreover, the constants a i satisfy a i −1 |E ij | a j for all i / j.
Proof.Let γ i : D × D → C be the bilinear map defined by i 1, . . ., M N γ i f, g β 1 fE ii , gE ii .

3.14
Since E ii is even, we have that γ i is a 2-cocycle in D.
The following statement was proved in 18 see Proof of Theorem 2.1 in page 74 and 3.2 and 3.3 in this work : if a 2-cocycle

3.15
Then β 1 aψ D for some a ∈ C. Now, using 3.4 , we have that γ i satisfies 3.15 ; thus, we get γ i a i ψ D for some a i ∈ C, proving the first part of this lemma.
In order to prove the second part, consider i / j.Then

3.16
Similarly, 3.17 Therefore, Proof.Since i / j, Proof of Theorem 2.1.From the previous lemmas, one can easily see that β 1 a 1 Ψ, by observing that the relation between the a i s in Lemma 3.3 is essentially the supertrace term in expression 2.12 of Ψ.
associative superalgebra of M N × M N super matrices with entries in D as .The Z 2 -gradation is the one inherited by the corresponding Z 2 -gradation in Mat M | N .By taking the usual superbracket we make SD M|N as into a Lie superalgebra, which is denoted by SD M|N .A set of generators is given by {t

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Lemma 3.1.For any f, g ∈ D as , β 1 fE ii , gE sj 0 if i / j or i / s.
Proof.Case j i and s / i.
t −1 E jj , which means that, a i −1 |E ij | a j for all i / j, finishing the proof.
ij , E ji for i / j and g ∈ D as .