Advances in Mathematical PhysicsVolume 2011 (2011), Article ID 870613, 9 pagesdoi:10.1155/2011/870613

Research Article

The Central Extension Defining the Super Matrix Generalization of

Carina Boyallian and Jose I. Liberati

Famaf-CIEM, Universidad Nacional de Córdoba, Ciudad Universitaria, 5000 Cordoba, Argentina

Received 13 June 2011; Accepted 1 August 2011

Academic Editor: Andrei D. Mironov

Copyright © 2011 Carina Boyallian and Jose I. Liberati. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract.

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

1. Introduction

The 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 which is the central extension of the Lie algebra of regular differential operators on the circle [1–5], and it contains the algebra as a subalgebra. Various extensions where constructed: super extension [6, 7], matrix version of [8], 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- 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 corresponds to . The general study of representation theory of 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 is defined as a specific central extension of the Lie superalgebra of regular differential operators on the superspace . Only in the special case of (i.e., ) 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 this result to the super matrix generalization . Similar studies of central extensions for -analogs and other versions can be found in [19, 20].

2. Basic Definitions and Main Result

Let and be two Lie superalgebras over . The Lie superalgebra is said to be a one-dimensional central extension of if is the direct sum of and as vector spaces and the Lie superbracket in is given by for all , where is the Lie bracket in and is a 2-cocycle on , that is, a bilinear -valued form satisfying the following conditions for all homogeneous elements : where denote the parity of . A central extension is trivial if is the direct sum of a subalgebra and as Lie algebras, where is isomorphic to . A 2-cocycle corresponding to a trivial central extension is called a 2-coboundary, and it is given by an as follows: for . It is easy to check that is a 2-cocycle. We say that the 2-cocycles are equivalent if is a 2-coboundary. The second cohomology group of with coefficients in is the set of equivalent classes of 2-cocycles, and it will be denoted by . If dim , we say that has an essentially unique nontrivial one-dimensional central extension.

Now, we will introduce the Lie superalgebra that will be considered in this work. Let us denote by Mat the associative superalgebra of linear transformations on the complex -dimensional superspace . Namely, we consider the set of all matrices of the form where are matrices, respectively, with complex entries. The -gradation is defined by declaring that matrices of the form (2.4) with are even, and those with are odd. We denote by the degree of with respect to this -gradation. The supertrace is defined by and it satisfies .

Let be the associative algebra of regular differential operators on the circle, that is, the operators on of the form The elements form its basis, where denotes . Another basis of is where . It is easy to see that Here and further we use the notation

Denote by the associative superalgebra of (super)matrices with entries in . The -gradation is the one inherited by the corresponding -gradation in Mat. By taking the usual superbracket we make into a Lie superalgebra, which is denoted by . A set of generators is given by .

Let be the central extension of by a one-dimensional vector space with a specified generator , whose commutation relation for homogeneous elements is given by where the 2-cocycle is given by

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

Theorem 2.1. One has the following: dim .

3. Proof of Theorem 2.1

We will need the explicit expression of the bracket of basis elements of type (2.9) in : In particular, we have

Let be a 2-cocycle on . We consider the linear functional in defined by Then is a 2-cocycle on that is equivalent to , and using (3.3), we obtain

In order to complete the proof we need to show that for some . By observing the supertrace that appears in the expression of in (2.12), we immediately obtain that for any In Lemmas 3.1 and 3.2, we will show that also satisfies (3.5).

Lemma 3.1. For any , if or .

Proof. Case and .Using that is even, , and (2.2), we obtain that where is the product in . Case and .In this case we have Case and .By taking the usual bracket, we make the associative algebra into a Lie algebra which is denoted by . Observe that It is easy to show that ; therefore, for any , we have Thus, if and , using (2.2), The proof is finished.

Lemma 3.2. For any and , when or .

Proof. If and , we have Hence we have .Finally, using skew-symmetry and the previous case, if , , and , we have that .

Now, it remains to consider the expression . In order to do it, consider again the Lie algebra (see (3.8)) and denote by the 2-cocycle defined in (2.12) with and .

In fact, from the expression of , we have

Lemma 3.3. There exist such that for all Moreover, the constants satisfy for all .

Proof. Let be the bilinear map defined by Since is even, we have that is a 2-cocycle in .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 in satisfies Then for some . Now, using (3.4), we have that satisfies (3.15); thus, we get for some , proving the first part of this lemma.In order to prove the second part, consider . Then Similarly, Therefore, , which means that, for all , finishing the proof.

Lemma 3.4. for and .

Proof. Since ,

Lemma 3.5. for and any .

Proof. Observe that Similarly, Hence, from (3.19) and (3.20), we obtain Since , we have . Therefore, (3.19) becomes the statement of this lemma.

Proof of Theorem 2.1. From the previous lemmas, one can easily see that , by observing that the relation between the in Lemma 3.3 is essentially the supertrace term in expression (2.12) of .

Acknowledgments

C. Boyallian and J.L. Liberati were supported in part by grants of Conicet, ANPCyT, Fundación Antorchas, Agencia Cba Ciencia, Secyt-UNC, and Fomec (Argentina).

References

H. Awata, M. Fukuma, Y. Matsuo, and S. Odake, “Representation theory of algebra,” in Proceedings of the Workshop Quantum Field Theory, Integrable Models and Beyond, T. Inami, et al., Ed., Yukawa Institute for Theoretical Physics, Kyoto University, Kyoto, Japan, February 1994.

H. Awata, M. Fukuma, Y. Matsuo, and S. Odake, “Representation theory of algebra,” Progress of Theoretical Physics Supplement, vol. 118, pp. 343–373, 1995.

I. Bakas, B. Khesin, and E. Kiritsis, “The logarithm of the derivative operator and higher spin algebras of type,” Communications in Mathematical Physics, vol. 151, no. 2, pp. 233–243, 1993.

V. Kac and A. Radul, “Quasifinite highest weight modules over the Lie algebra of differential operators on the circle,” Communications in Mathematical Physics, vol. 157, no. 3, pp. 429–457, 1993.

C. N. Pope, L. J. Romans, and X. Shen, “Ideals of Kac-Moody algebras and realisations of ,” Physics Letters B, vol. 245, no. 1, pp. 72–78, 1990.

H. Awata, M. Fukuma, Y. Matsuo, and S. Odake, “Quasifinite highest weight modules over the super algebra,” Communications in Mathematical Physics, vol. 170, no. 1, pp. 151–179, 1995.

E. Bergshoeff, M. Vasiliev, and B. de Wit, “The super- algebra,” Physics Letters B, vol. 256, no. 2, pp. 199–205, 1991.

S. Odake and T. Sano, “ and super- algebras with SU(N) symmetry,” Physics Letters B, vol. 258, no. 3-4, pp. 369–374, 1991.

S. Odake, “Unitary representations of W infinity algebras,” International Journal of Modern Physics A, vol. 7, no. 25, pp. 6339–6355, 1992.

L. Guieu and C. Roger, L'Algèbre et le Groupe de Virasoro, Aspects Géométriques et Algébriques, Généralisations, Les Publications CRM, Montreal, Canda, 2007.

A. O. Radul, “Lie algebras of differential operators, their central extensions and W-algebras,” Functional Analysis and Its Applications, vol. 25, no. 1, pp. 25–39, 1991.

C. Boyallian, V. G. Kac, J. I. Liberati, and C. H. Yan, “Quasifinite highest weight modules over the Lie algebra of matrix differential operators on the circle,” Journal of Mathematical Physics, vol. 39, no. 5, pp. 2910–2928, 1998.

S.-J. Cheng and W. Wang, “Lie subalgebras of differential operators on the super circle,” Research Institute for Mathematical Sciences Publications, vol. 39, no. 3, pp. 545–600, 2003.

V. G. Kac and J. I. Liberati, “Unitary quasi-finite representations of ,” Letters in Mathematical Physics, vol. 53, no. 1, pp. 11–27, 2000.

C. Boyallian, V. G. Kac, and J. I. Liberati, “On the classification of subalgebras CendN and gcN,” Journal of Algebra, vol. 260, no. 1, pp. 32–63, 2003.

V. Kac, Vertex Algebras for Beginners, vol. 10 of University Lecture Series, American Mathematical Society, Providence, RI, USA, 2nd edition, 1998.

V. G. Kac, “Formal distribution algebras and conformal algebras,” in XIIth International Congress of Mathematical Physics (ICMP '97) (Brisbane), pp. 80–97, Internat Press, Cambridge, Mass, USA, 1999.

W. L. Li, “2-cocyles on the algebra of differential operators,” Journal of Algebra, vol. 122, no. 1, pp. 64–80, 1989.

W. Li and R. L. Wilson, “Central extensions of some Lie algebras,” Proceedings of the American Mathematical Society, vol. 126, no. 9, pp. 2569–2577, 1998.

D. Liu and N. Hu, “Derivation algebras and 2-cocycles of the algebras of q-differential operators,” Communications in Algebra, vol. 32, no. 11, pp. 4387–4413, 2004.