ISRN Mathematical PhysicsVolume 2013 (2013), Article ID 672872, 13 pageshttp://dx.doi.org/10.1155/2013/672872

Research Article

Quasifinite Representations of Classical Subalgebras of the Lie Superalgebra of Quantum Pseudodifferential Operators

José I. García and José I. Liberati

Famaf-Ciem, Universidad Nacional de Córdoba, Ciudad Universitaria, 5000 Córdoba, Argentina

Received 11 June 2013; Accepted 5 July 2013

Academic Editors: A. M. Gavrilik, M. Martins, and K. Netocny

Copyright © 2013 José I. García and José 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 classify the anti-involutions of the superalgebra of quantum pseudodifferential operators on the super circle preserving the principal gradation, producing in this way a family of Lie subalgebras minus fixed by these anti-involutions. We classify the irreducible quasifinite highest weight representations of the central extension of these Lie subalgebras.

1. Introduction

The -infinity algebras naturally arise in various physical theories, such as conformal field theory and the theory of quantum Hall effect (see [1, 2] and references therein). The algebra, which is the central extension of the Lie algebra of differential operators on the circle, is the most fundamental among these algebras.

The difficulty in understanding the representation theory of a Lie algebra of this kind is that although admits a natural -gradation, each of the graded subspaces is still infinite dimensional in contrast to the more familiar cases such as the Virasoro algebra and Kac-Moody algebras. Therefore, the study of the highest weight modules which satisfy the quasifiniteness condition, that its graded subspaces have finite dimension, becomes a nontrivial problem. The systematic study of quasifinite highest weight modules of was initiated by Kac and Radul in [2] and further studied in [1, 3–5] and many others.

By analyzing for which parabolic subalgebras of   the corresponding generalized Verma modules are quasifinite, Kac and Radul [2] gave a characterization of quasifinite highest weight -modules in terms of certain generating function of highest weights and these modules where constructed in terms of irreducible highest weight representations of the Lie algebra of infinite matrices.

The classification and construction of quasifinite modules for the matrix version (denoted by ), super analog, -analog, and super -analog of , were developed in [1, 2, 4, 6], respectively.

The Lie algebra , recently studied in [4], correspond to the central extension of the algebra of matrix differential operators on the circle. The study of the representation theory of some interesting subalgebras of , and its -analog and super version [1], is not complete.

Another important example is the Lie algebra which is a particular case of a family of subalgebras of , where () is the central extension of the Lie algebra of differential operators on the circle that are a multiple of .

This Lie algebra was studied by Kac and Liberati in [3]; observe that . Following the ideas of Kac-Radul [2], in [3] they obtained the classification of the irreducible quasifinite highest weight modules over . They also developed a general theory of quasifinite highest weight modules over -graded Lie algebras. These general results were extended to the super version in [6].

A natural source of subalgebras comes from the subalgebras minus fixed by an anti-involution of the corresponding associative algebra, which preserve the gradation. In [7], Bloch finds an anti-involution of and he shows a relation between the representations of the corresponding subalgebra and certain values of the Riemann zeta function. In several and recent papers ([5, 8–10]) the authors obtained the classification of the anti-involutions of certain algebras and then they characterize the irreducible quasifinite highest weight modules of the corresponding subalgebras minus fixed by these anti-involutions, obtaining orthogonal and symplectic subalgebras of , , and so forth.

The main goal of this work is to present a -analog of Cheng-Wang [8] or a super-analog of Boyallian-Liberati [9]; namely, we classify the anti-involutions of the superalgebra that preserve the gradation, where - is the superalgebra of regular pseudodifferential operators on the super circle . Then we present the classification of the irreducible quasifinite highest weight modules over the Lie subalgebras minus fixed by these anti-involutions.

Table 1 describes the map on the classification of irreducible quasifinite highest weight modules over , the matrix version, super analog and -analog, and for the subalgebras constructed from the anti-involutions, showing the place of the present results in this long-term program.

Table 1

super
q-
super q-

Algebra
[2]
[3, 11]
[4]
[1]
[2]
[6]

Anti-involution
[5]
[10] (particular case in [12])
Partial results in [13] and others
[8]
[9]
Present work

The work is organized as follows. In Section 2, we classify the anti-involutions of - that preserve the gradation, where is the superalgebra of regular pseudodifferential operators on the super circle . In Section 3, we recall the general results on quasifinite modules over a graded Lie superalgebra and we present the classification of the irreducible quasifinite highest weight modules over the Lie subalgebras minus fixed by the anti-involutions obtained in Section 2.

2. Anti-Involution of Preserving Its Principal Gradation

Let with . Now, denotes the following operator on : We denote by the associative algebra of all pseudodifferential operators, that is, the operators on of the form Any pseudodifferential operator can be written as linear combinations of elements of the form , where and . The product in is given by Letting , we define the principal -gradation of .

Moreover, we denote by the set of supermatrices with coefficients in , viewed as the associative superalgebra of linear transformations of the complex -dimensional superspace . And we denote by the matrix with in the -place and everywhere else. Declaring , even and , odd elements, we endow with a -gradation where denotes the parity of the homogeneous element .

We denote by the associative superalgebra of supermatrices with entries in , namely, and the product is given by the usual matrix multiplication. Let denote the Lie superalgebra obtained from by taking the usual bracket.

Now, we introduce the linear map as where if , and we define the 2-cocycle in by Then we denote by the one-dimensional central extension of with central charge corresponding to the 2-cocycle , namely, , where the bracket is given by

Letting ,    and , where ,  , defines the principal -gradation of , and . This equips , , and with -gradations compatible with their -gradation; thus, where

An anti-involution of is an involutive anti-automorphism of ; that is, with ,   and  , where , and .

The main result of this section is the following theorem with the classification of all anti-involutions of that preserve the principal -gradation.

Theorem 1. Any anti-involution of which preserves the principal gradation is one of the following (, ): (a) with , such that ; (b) with , such that and .

We divide the proof of Theorem 1 into several results.

Let be an anti-involution of which preserves the principal gradation; then, defines linear maps preserving the principal gradation

Lemma 2. Let be an anti-involution of preserving the principal gradation. Then satisfies one of the following conditions: (a)(b) For some , where , and .

Proof. Since preserves the principal gradation, we have that for some , with , and .Then, since , we have two possibilities or Moreover, since , for , , from (15) we have that Finally, since the result follows from (16), (17), (18), and (19).

Corollary 3. Let be an anti-involution of which preserves the principal gradation. Then one has the following. (a)If satisfies Lemma 2(a), then , for all , . (b)If satisfies Lemma 2(b), then , for all , .

Proof. It is clear that ; then the result follows from Lemma 2.

Proposition 4. Let be an anti-involution of which preserves the principal gradation and let one assume that it satisfies Lemma 2(a). Then is one of the from Theorem 1(a).

In order to prove Proposition 4 we will need the following result.

Lemma 5. Let be as in Proposition 4; then, where , such that , with .

Proof. It is easy to check that is a anti-involution of with ; then, from Section 3 in [9] and hypothesis, we obtain that with , such that , . Besides, from Corollary 3(a) and hypothesis, we have that Therefore and with , ; then, Moreover, with , . The proof follows from (21) and by replacing (21) and (23) in (24).

Proof of Proposition 4. From Lemma 5, we have that Then from (25)-(26), we obtain that and also from (26)-(27) we obtain Then, from (27) and again using Lemma 5, we have that for all . Comparing (29) with (30), we obtain that in particular, From (28) and replacing (27) and (32) in (20), we obtain that , finishing the proof.

Proposition 6. Let be an anti-involution of which preserves the principal graduation, and let one assume that it satisfies Lemma 2(b). Then is one of the from Theorem 1(b).

In order to prove Proposition 6 we will need the following result.

Lemma 7. Let be as in Proposition 6, then where , , and .

Proof. From hypothesis and Corollary 3(b), we have that Then, using (34), we have that and by induction we obtain that where , , ,  . Then, from (37) we have that with , .On the other hand, from (35) we have that and with and ; therefore, Moreover, with . The proof follows from (38) and by replacing (38) and (39) in (40).

Proof of Proposition 6. By Lemma 7, we have that Then, using (41), we have that where , . Suppose that ; then, from (45) and (47) we have that , , and by replacing them in (46), we obtain that , but since is not a root of unity, we necessarily have that . Then, using Lemma 7 we get moreover, Comparing (49) with (50), we obtain that which is a contradiction. Therefore, by (44) and the aforementioned result, we get that Then replacing (51) in (45) and (47) we obtain that On the other hand, comparing (42) with (43), we have that and by replacing (51), (52), and (53) in (46), we obtain that Using (48) and (54) and by replacing (51), (52), and (53) in (33), we obtain that , finishing the proof.

Proof of Theorem 1. It is straightforward to check that the two cases are anti-involutions. Reciprocally, from Lemma 2 and Propositions 4 and 6, it is clear that any anti-involution of which preserves the principal gradation satisfies (a) or (b), finishing the proof.

Now, given an anti-involution of , one can check that the set of points minus -fixed is a subalgebra of . Moreover, if preserves the principal -graduation, this subalgebra inherits the -graduation. These subalgebras are described in the last part of this section.

We define the following automorphisms of by with . On the other hand, given , we denote

Remark 8. We see that with if and only if , for all .

The Case . Let , , be such that . We denote by the Lie subalgebra of consisting of -fixed points; then, it inherits a -gradation from ; therefore, , where Moreover, we denote and .

The following lemma gives a description of .

Lemma 9. Let , , , be as aforementioned; then, and for all .

Proof. First, it is easy to check that Then if we take such that and , from (59) and the relation between , we obtain that .On the other hand, by Theorem 1(a) and linearity of we obtain that (for ) if and only if , . Now, let be with ; then, if and only if , for all , which is equivalent to (see Remark 8). Therefore, Similarly, we prove that if and only if Now, we suppose that ; then, if and only if , finishing the proof.

The Case . Let , , be such that and . We denote by the Lie subalgebra of consisting of -fixed; then, it inherits a -gradation from ; thus , where Moreover, we denote , and , , with . The following lemma gives a description of . We will need the following notation:

Lemma 10. Let , , , , be as aforementioned; then, is isomorphic to some of the following algebras: , , , or . And for all .

Proof. It is easy to check that, then; if we take such that , we obtain the first assertion using (67) and the relations between .On the other hand, we suppose that ; then, if and only if . The proof for is similar.Now, by Theorem 1(b) and linearity of , we obtain that if and only if and are elements in . We suppose that , with ; then, if and only if  , for all , which is equivalent to (see Remark 8). Therefore, Similarly, we prove that if and only if , with .The other proofs are similar.

Remark 11. From Lemma 9 (resp., Lemma 10), it is clear that an element in (resp., ) is totally determined by its value in the position (resp., ), where .

Finally, we denote by and the central extensions of and corresponding to the restrictions of the 2-cocycle , respectively. It is clear that these subalgebras admit a -graduation compatible with their -graduation; that is, where with and , for all .

3. Quasifinite Highest Weight Modules over and

The goal of this section is to characterize the quasifinite irreducible highest weight modules over and ; for this we will apply the general results on quasifinite representations of -graded Lie superalgebras developed in Section 2 in [6]. Let us recall some general definition and results from [6].

In this section, denote a consistent -graded Lie superalgebra over ; namely, and also We denote . A subalgebra of is called parabolic if We assume the following properties on :(SP1) is commutative,(SP2) if () and , then .Given that is nonzero, we define , where It was proved in [6] that is the minimal parabolic subalgebra containing and also that .

Definition 12. (a) A parabolic subalgebra is called nondegenerate if has finite codimension in , for all . (b) An element is called nondegenerate if is nondegenerate.

We will also require the following condition on .(SP3) If is a nondegenerate parabolic subalgebra of , then there exists a nondegenerate element .

A -module is called quasifinite if for all . Given , a highest weight module is a -graded -module , defined by the following properties: (a), where is a nonzero vector; (b), for all ; (c); (d).

A nonzero vector is called singular if .

The Verma module over is defined as usual: where is the one-dimensional ()-module given by if , and and the action of on is induced by the left multiplication in . Any highest weight module is a quotient module of . The irreducible module is the quotient of by the maximal proper graded submodule.

Now, let be a parabolic subalgebra of and let be such that . Then the ()-module extends to a -module by letting for all , and we may construct the highest weight module called the generalized Verma module. Clearly all these highest weight modules are graded. The following result gives the characterization of all irreducible quasifinite highest-weight modules.

Theorem 13. Let be a consistent -graded Lie superalgebra over that satisfies conditions (SP1), (SP2), and (SP3). The following conditions on are equivalent. (a) contains a singular vector in , where is nondegenerate. (b)There exists a nondegenerate element , such that . (c) is quasifinite. (d)There exists a nondegenerate element , such that is the irreducible quotient of the generalized Verma module .

Proof. See [6].

Moreover, we will need the following result. Recall that a quasipolynomial is a combination of functions of the form , where is a polynomial and . That is, it satisfies a nontrivial linear differential equation with constant coefficients.

Proposition 14. Given a quasipolynomial and a polynomial , take where ; then, if and only if .

Below, we prove that and satisfy the properties (SP1), (SP2), and (SP3), which is equivalent to study of its parabolic subalgebras. Then using Theorem 13 and Proposition 14, we obtain two equivalent characterizations of the quasifinite highest weight modules of these algebras. Before studying each particular case, we will consider the following useful result.

Lemma 15. Let and be nonzero elements in , where ,   and let be nonconstant. Then one has the following. (a) or , with . (b)If , then or .

Proof. (a) We suppose that and ; then, Using (79) and the hypothesis, we have that and are nonzero. Then, if we replace (79) with (80), we obtain which is a contradiction since is not constant, , and is not unity root. The other case is similar.The proof of (b) is similar to the proof of (a).

3.1. The Case

It is clear that satisfies (SP1). Now, we suppose that (with ) satisfies ; in particular for all Then using (81), we obtain that for all . Hence, (since and is not a root of unity); therefore, . Now, we suppose that in ; (with ) satisfies then, for all we have that which is equivalent to for all . Hence ; therefore, . Thus, we prove that satisfies the property (SP2).

Finally, using the following lemma, we will prove that satisfies the property (SP3).

Lemma 16. Let be a -graded subalgebra of , where . Then one has the following. (a)For each , has finite codimension in if and only if . (b)For each , has finite codimension in if and only if there exists , such that and are nonzero. (c) has finite codimension in , for all if and only if .

Proof. (a) We suppose that there exists with ; then and belong to , for all . Moreover, and for all . Therefore, using (84), we have that for all , where is the ideal of generated by . Using that has finite codimension in , we obtain that has finite codimension in (see Remark 11). The proof of the converse is trivial. (b) We suppose that , with nonzero and in ; then similar to the aforementioned argument where , and this is true for all . Therefore Since and have finite codimension in , we obtain that has finite codimension in . The proof of the converse is trivial. (c) We suppose that ; then, in order to prove that has finite codimension in , for all , we only need to see that this is true for all , since by using (SP2), we obtain that for all ; then, from (a) we have that has finite codimension in for all . By hypothesis, there exists with ; then, using (84), we have that , and where and are nonzero. Hence, by (b), has finite codimension in . Moreover, by (87) we obtain that for all . Now, by induction we suppose that has finite codimension in ; then, there exists where and are nonzero, and by (89) and Lemma 15(b), for some no constant in ; therefore, there exists a nonzero element in . Similarly, using (90) and Lemma 15(b), we see that there exists a nonzero element in ; then, using (b), has finite codimension in , finishing the induction. The proof of the converse is trivial.

Corollary 17. (a) Any parabolic subalgebra of is nondegenerate. (b) Any nonzero element of is nondegenerate. (c) satisfies (SP3).

Proof. Let be a parabolic subalgebra of ; by definition there exists such that , and then by (SP2) ; the proof of (a) follows from Lemma 16(c). Finally, (b) follows from (a), and (c) follows from (b).

A functional is described by its labels , with , and the central charge . We will consider the generating series

Theorem 18. An irreducible highest weight -module is quasifinite if and only if one of the following equivalent conditions holds. (a)There exists a Laurent polynomial such that (b)There exists a quasipolynomial such that for all .

Proof. By Theorem 13, is quasifinite if and only if there exist a nondegenerate element in , such that .Now, let be a nonzero element of (with ); then, by Corollary 17(b), is a nondegenerate element. Then, if and only if, for all , Multiplying (95) by and adding over , we obtain that The equivalence between (a) and (b) follows from Proposition 14.

3.2. The Case

It is clear that satisfies (SP1). Now, we prove that satisfies (SP2). We suppose that in (with ) satisfies ; in particular if we take nonzero elements and in , we have that that is, ; therefore, . Similarly, which is equivalent to ; therefore, , and then .

Remark 19. Given and with , then , where and

Now, let (with ) be such that is nonzero; then, if we take nonzero in by Remark 19, there exists such that is not constant; then by Lemma 15(a), we have that Therefore satisfies (SP2).

In order to prove that satisfies (SP3), we will need the following result. We denote

Lemma 20. Let be a -graded subalgebra of , with . Then one has the following: (a)for each ,   has finite codimension in if and only if ; (b)for each ,   has finite codimension in if and only if there exists , such that and are nonzero; (c) has finite codimension in for all , if and only if has finite codimension in .

Proof. (a) Let , with ; then, and for all ; therefore, by (103) we have that for all , and since has finite codimension in , we have that has finite codimension in (see Remark 11). The proof of the converse is trivial. (b) We suppose that there exist nonzero elements then, using Remark 19, we obtain that for all (see (99)) and for all (see (100)). Then using that has finite codimension in and has finite codimension in , we obtain that has finite codimension in . Therefore in order to prove that has finite codimension in , we only need to see that there exist nonzero elements as aforementioned in . Let with nonzero and , and let and be as in Remark 19. Then, taking (observe that ), we obtain that Similarly, taking (observe that ), we obtain proving the existence of nonzero elements of those forms. The proof of the converse is trivial. (c) We suppose that has finite codimension in ; then, in order to prove that has finite codimension in for all , we only need to see that this is true for all with , since in this case by using (SP2), we obtain that for all ; then, by (a), we have that has finite codimension in , for all .By hypothesis, there exist nonzero elements and in ; therefore, is nonzero. By induction, we suppose that has finite codimension in ; then, there exist nonzero elements and in , and by Remark 19, we also have , in with no constant for all . Then by Lemma 15(a), we have that Therefore from (108), there exists a nonzero element in . Similarly, we prove that there exists nonzero in ; finally the proof follows from (b). The proof of the converse is trivial.

Corollary 21. (a) is nondegenerate if and only if and are nonzero. (b) satisfies (SP3).

Proof. (a) Let . Then, if and are nonzero by Lemma 20(b) and has finite codimension in , then by Lemma 20(c), is a nondegenerate element. Reciprocally, if or , then by definition . (b) Let be a nondegenerate parabolic subalgebra of ; then there exists where and are nonzero. Then the proof follows from (a).

A functional is described by its labels, with and the central charge . Moreover, we define . We consider the generating series

Theorem 22. An irreducible highest weight -module is quasifinite if and only if one of the following equivalent conditions holds. (a)There exist Laurent polynomials and such that (b)There exist quasipolynomials and such that for all .

Proof. By Theorem 13, is quasifinite if and only if there exists a nondegenerate element in , such that .Now, let , with and such that and are nonzero. Then by Corollary 21(a), is a nondegenerate element. Then, if and only if, for all , From (112), we obtain that Multiplying (114) by and adding over , we obtain that Then, by (113) and using Remark 8, we obtain that Multiplying (116) by and adding over , we obtain that The equivalence between (a) and (b) follows from Proposition 14.

Acknowledgments

The authors were supported in part by grants of Conicet, Foncyt-ANPCyT, and Secyt-UNC (Argentina).

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