Lie Subalgebras of the Matrix Quantum Pseudodifferential Operators

Abstract

We give a complete description of the anti-involutions that preserve the principal gradation of the algebra of matrix quantum pseudodifferential operators and we describe the Lie subalgebras of their minus fixed points.

1. Introduction

The -infinity algebras naturally arise in various physical theories, such as conformal field theory and the theory of quantum Hall effect. 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 representations of Lie algebra were first studied in [1], where its irreducible quasifinite highest weight representations were characterized. At the end of that article, similar results were found for the central extension of the Lie algebra of quantum pseudodifferential operators , which contains as a subalgebra the q-analogue of the Lie algebra , the algebra of all regular difference operators on . Here and further, is not a root of unity.

In [2], certain subalgebras of the Lie algebra were considered, and it was shown that there are, up to conjugation, two anti-involutions on , which preserve the principal gradation. These results were extended to the matrix case in [3], where a complete description of the anti-involutions of the algebra of the -matrix differential operators on the circle preserving the principal -gradation was given.

Analogously, in [4] it was shown that there is a family of anti-involutions on   , up to conjugation, preserving the principal gradation. The goal of this paper is to extend these results to the matrix case, where the global image seems to be richer and more complex.

The paper is organized as follows: in Section 2 we give a complete description of the anti-involutions of the algebra of -quantum pseudodifferential operators, preserving the principal -gradation. For each with , we obtain, up to conjugation, two families of anti-involutions that show quite different results when and . To exhibit their differences in detail, they are studied separately in Sections 3 and 4, respectively. In Section 3, the anti-involutions give us two families of Lie subalgebras fixed by . Then, we give a geometric realization of , concluding that is a subalgebra of of orthogonal type and is a subalgebra of of symplectic type. In Section 4, the families , with , give us two families of Lie subalgebras    fixed by . We give a geometric realization of , concluding that is a subalgebra of of type and is a subalgebra of of type .

2. Quantum Pseudodifferential Operators

Consider the Laurent polynomial algebra in one variable. We denote by the associative algebra of quantum pseudodifferential operators. Explicitly, let denote the operator on given bywhere . An element of can be written as a linear combination of operators of the form , where is a Laurent polynomial in . The product in is given by

Now let denote the Lie algebra obtained from by taking the usual commutator. Let . It follows that

Let be a positive integer. As of this point, we shall denote by the associative algebra of all -matrices over an algebra and by the standard basis of .

Let be the associative algebra of all quantum matrix pseudodifferential operators, namely, the operators on of the form

In a more useful notation, we write the pseudodifferential operators as linear combinations of elements of the form , where is a Laurent polynomial, and . The product in is given by

Let denote the Lie algebra obtained from with the bracket given by the commutator; namely,

The elements form a basis of .

Define the weight on by

This gives the principal -gradation of and , the latter of which is given by . This allows the following triangular decomposition:where and .

An anti-involution of is an involutive antiautomorphism of ; that is, , , and , for all and . From now on we will assume that .

As we intend to classify the anti-involutions of preserving its principal gradation, we shall introduce some notation. For each , define the permutation in by

Let us fix ,   and , , and write

We define in by the following formulas:

Theorem 1. Let defined on generators by (11) extends to an anti-involution on which preserves the principal -gradation if and only ifMoreover, any anti-involution of which preserves the principal -gradation is of the form .

The proof will mainly consist of several steps making use of the involutive property of and the relations between the generators .

Proof. Fix .
Step 1. Because should preserve the principal -gradation, we have . Given the fact that is an anti-involution, we get , so . Taking into consideration the positive and negative degrees of these Laurent polynomials, we arrive at , where are constant elements such that . This gives us or for every . We also know that . So, and for . So, for each there exists a unique such that and for any . And for every and . In particular, for , so for any , obtaining that . Due to the injectivity of , is a permutation in , and since is an involution, we have .
Step 2. Again, due to the fact that should preserve the principal -gradation, we may assume that and . So,Proceeding similarly with , we have Combining these two equations, we have So, and, as consequence, they must be units of the Laurent polynomial ring. Therefore, we can assume and , with and . So, is then determined by .
Now, let us note that we can write , for every . Therefore,So, and . This gives us the following alternatives or .
Step 3. Since and should preserve the principal -gradation, we can assume and . Using a similar argument to the one used in Step 2 and denoting and , we can deduce that and for , and also and , with , , and . So,Therefore, we have and . On the other hand,We can therefore conclude that and . From this last equation and the previous step, we get .
If , then . Since we assumed that is not a root of unity, it is easy to check that these are not antiautomorphisms. Therefore, , and .
By now, whereand also and , for , and .
Step 4. Suppose . As an implication of the -gradation preservation property of , we have that Since , we can deducewhere and .
Similarly, if and we deducewhere .
Case 1. Let , with :using (21), we must have because we would otherwise get in the right hand side above, soThen, and are units of the Laurent polynomial ring and . Therefore, because , we can write and , withCase 2. Let and if , in the same way, using simultaneously (21) and (22) in order to take care of that appears in , we have ; thus,Therefore, and we can assume and , withCase 3. Let and :using (22), we must have in order to avoid getting in the right hand side above. SoThen, and and are units of the Laurent polynomial ring, so we can assume and , withCase 4. Let and if , since is an involution, we make use of (21) and (22) simultaneously to take care of appearing in . In order to do this, we require . SoThen, . Once again, with and being units of the Laurent polynomial ring, we can write and , withStep 5. Let ; then, by Step 1, . Using (21) with condition , we have that ; therefore, trivially. So,with We can finally rewrite if and .
Now, in the case and in (21), we have and it is immediate that ; then,So, Because of this, we have if and .
Thus, we can rewrite (21) and (22) as the following: for and for We now intend to determine the permutation . So, let be such that . In Cases 1 and 3, and it is easy to see thatMoreover, since in Case 2 , we have . Since is a bijective map, we conclude that must be given in (9) where .
Let us note that if , , and if , . As a consequence, we can easily see that if (Case 1) corresponds to the choice or , and the case in which and corresponds to (Case 2). Similarly for , when or , we have (Case 3) and the case in which and corresponds to (Case 4).
Computing in the four cases for and , with their corresponding restrictions, we have the following.
In Case 1, where and or , we getRegarding (41), when we deduce, combining (41) with (25), that .
On the other hand, from in (41) combined with the fact that , we get . So,
Now, due to (25) and (40), with and . So,
If we consider and in (40) and (25), we have . Using (20) and the fact that , we get . So, Thus,resembling [4], andIn Case 2, where and , we haveRegarding (45), when , we deduce thatOn the other hand, when in (45),  . Combining the last two items, we get .
Now, due to (44) with and , Combining this with (44), we get that, for arbitrary values of and ,If we consider and in the last equation, we get . So, and due to (42) and (43), .
Finally, when and in (47), is constant for every . So, for every .
Now, because of (46),Letting and in (47), we have . Since , , resulting in because of (42). So, and, by (48), and in (42) and (43), this implies .
Again, letting and in (47), and combining this with the previous equation, we get .
Cases 3 and 4 give the same results.
We have thus arrived at the final relations of (11).
Now, recall that we have, for . So, rewriting (37) and (38) for these cases, we haveIf , since , we get (12a). Finally, (12b) are results of (25) and of (44) with and and taking into consideration that and .
On the other hand, it is straightforward to check that defined by (11) is indeed anti-involution of , finishing the proof.

Corollary 2. If , the anti-involution is given bywhere and verify relations (12a) and (12b).

Proof. If there is only Case 1 to be considered in the proof of Theorem 1.

Remark 3. Case coincides with [4].
We will now concentrate on the implications of conditions (12a) and (12b). First, let us note that, as a consequence of (12a), all coefficients are completely determined byand the upper condition of (12b) can be written as by (39). Combining the lower condition of (12b) with (12a), we get . Also, let us note that the permutation is given by two simple permutations of the sets and . Thus, (12b) reduces toLet and let us analyze the previous formulas. If (resp., ) is even, by (52) we have and (resp., and ). The coefficient (resp., ) will be called a fixed point.
Case −. If is even and (1)is even, condition (53) is satisfied if there are two fixed points: one of them must be and the other one must be equal to ,(2) is odd, then there are no fixed points and (53) is impossible. Thus, there is no anti-involution in this case.If is odd, then or is even and we have only one fixed point that must be equal to .
Case +. For any , condition (53) is satisfied if the (possible) fixed points are all equal to .

From now on, we will consider separately cases and in an attempt to exhibit more clearly their particular results.

3. Case

3.1. Lie Subalgebras of

Let denote the Lie subalgebra of fixed by minus ; namely,where , for , is given by

Note that from [4] agrees with for .

Let us now analyze the relation among for different values of , , , , and . To that end, let and denote by the automorphism of given by , , and , where and stands for the identity matrix. It is easy to check that preserves the principal -gradation of . Making use of the equation for pointed out in (55), we havewhich resembles [4], when .

Similarly, let satisfying (12a) and (12b). Denote by the automorphism of defined by ,  , andLet ; then, we havewhere and . Observe that and also satisfy (12a) and (12b). Using (56) and (58), we have the following.

Lemma 4. The Lie algebras for arbitrary choices of , , and are isomorphic to , where is or and is the matrix with except for the fixed points that are or , which keep their sign.

We shall introduce some notation in order to give an explicit description of this family of subalgebras.

First, we will write and instead of and . Also, for any matrix , definethat is, the transpose with respect to the “other” diagonal. Recall the anti-involutions on given in [4]:

An extension of to a map on can be made by taking .

Case +. We define the following map on :

Explicitly, the anti-involution on is given bywhere and

Case −. Now, consider the following map on :

Then, on is explicitly given bywhere . And

Let us note that are Lie subalgebras of and that and are antiautomorphisms.

Remark 5. Replacing by (usual transpose) in (61) and (64) gives us another family of involutions that we shall denote by , which do not preserve the principal -gradation. Moreover, the corresponding subalgebras are not -graded subalgebras of , even though they are isomorphic to the others using , where is the following -matrix:This way, we get .

3.2. Generators of

We can now give a detailed description of the generators of .

Let us denote (where or ) the set of Laurent polynomials such that .

Note that and observe that by (60)

Here and in the following we will use the description of the elements in the subalgebras used in (63) and (66). The following is a set of generators of :and the generators on the opposite diagonal are

3.3. Geometric Realization of

In this subsection, we give a geometric realization of . The algebra acts on the space and we define two bilinear forms on :where , as in (67), and given by .

Proposition 6. (a) The bilinear forms are nondegenerate. Moreover, is symmetric and is antisymmetric.
(b) For any and we have where . In other words, and are adjoint operators with respect to .

Proof. (a) The statements are straightforward.
(b) Let , , and . Recall thatSo,On the other hand, we haveNote that if we multiply (74) by and (75) by , we getIt is easy to prove that, for ,Making use of this result in (76), we can see thatThus, as expected, we get

Remark 7. In a similar fashion, we can define the following nondegenerate bilinear forms on : wherewith the identity matrix, and it easily follows that they satisfywhere were defined in Remark 5.

4. Case

4.1. Lie Subalgebras of

Let denote the Lie subalgebra of fixed by minus :As in the case , we analyze the relation among for different values of , and . Let and denote by the automorphism of given by , , and , where stands for the identity matrix and . Clearly preserves the principal -gradation of . As before, we have for this case the following:

Let satisfying (12a) and (12b) and denote by the automorphism of defined by , , andLetting , we havewhere and . Note that and also satisfy (12a) and (12b). Making use of (84) and (86), we have the following.

Lemma 8. The Lie algebras for arbitrary choices of and are isomorphic to , where is or and is the matrix with except for the fixed points that are or , which keep their sign.

Remark 9. Due to this lemma, we can find a complex number such that . Moreover, recall that, in the case , we find a complex number such that (which makes as a consequence). So, in both cases, the subalgebra is isomorphic to and the only distinction between both cases is regarding : while takes an arbitrary value in the case , if happens to be .
We will write and instead of and , with and . As in the previous section, for any matrix , we definethat is, the transpose with respect to the “other” diagonal. Recall once again the anti-involutions on given in [4]:We extend to a map on by taking . Now let .
Case +. We define the following maps:where , , , and . We can write the anti-involution on explicitly asThe fact that implies and , and these two conditions are equivalent because . Moreover, proving that is a Lie subalgebra of by direct computations requires using the fact that and are antiautomorphisms and the identities , , , , and so forth. Observe, however, that and are not antiautomorphisms. The following identities are also useful:Case −. As we have seen in the analysis following equation (53), the case even and (also ) odd is impossible. Therefore, we may suppose, due to the symmetry, that is even. Now, we shall consider the following maps:where , , , and . Then, the anti-involution on is explicitly given byAs before, condition implies and , and these two conditions are equivalent due to the fact that . Moreover, proving that is a Lie subalgebra of by direct computations requires using the fact that and are antiautomorphisms and the identities , , , , and so forth. Once again, and are not antiautomorphisms. We also need to use

Remark 10. Replacing by (usual transpose) in (89) and (93) gives another family of involutions denoted by . These involutions do not preserve the principal -gradation, and the corresponding subalgebras are not -graded subalgebras of , but they are isomorphic to the others using the same argument in Remark 5.

4.2. Generators of

In this subsection we give a detailed description of the generators of .

Let us denote (where or ) the set of Laurent polynomials such that And let if is odd and if is even.

Recall thatand also

Therefore, the following is a set of generators of , using the description of the elements in the subalgebras given in (91) and (95).(i)For block , where ,and the generators on the opposite diagonal are (ii)For blocks (and ), where and (or and ), (iii)For block , where , and the generators on the opposite diagonal are

4.3. Geometric Realization of

In this section, we give a geometric realization of .

The algebra acts on the space and we define two bilinear forms on :wherewith given by , and as in (67). Observe that is the orthogonal decomposition of . Now, consider the following proposition.

Proposition 11. (a) The bilinear forms are nondegenerate. Moreover, is symmetric and is symmetric in the subspace and antisymmetric in .
(b) For any and , we have that is, and are adjoint operators with respect to .

Proof. (a) The statements are straightforward.
(b) Let , and be as shown previously. Recall thatSo,On the other hand, we haveAs the last two results are equal, we finish the proof.

Remark 12. In a similar fashion, we can define the following nondegenerate bilinear forms on : wherewith the identity matrix, and it easily follows that they satisfywhere were defined in (67).

Competing Interests

The authors declare that they have no competing interests.