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

Abstract

We classify the anti-involutions of the superalgebra of quantum pseudodifferential operators on the super circle 𝑆^1|1 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 𝒲-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 𝑊_1+∞ algebra, which is the central extension of the Lie algebra 𝒟 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 𝑊_1+∞ admits a natural ℤ-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 𝑊_1+∞ 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  𝑊_1+∞ the corresponding generalized Verma modules are quasifinite, Kac and Radul [2] gave a characterization of quasifinite highest weight 𝑊_1+∞-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 𝑊^𝑁_1+∞), super analog, 𝑞-analog, and super 𝑞-analog of 𝑊_1+∞, were developed in [1, 2, 4, 6], respectively.

The Lie algebra 𝑊^𝑁_1+∞, 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 𝑊^𝑁_1+∞, and its 𝑞-analog and super version [1], is not complete.

Another important example is the Lie algebra 𝑊_∞ which is a particular case of a family of subalgebras 𝑊_∞,𝑝 of 𝑊_1+∞, where 𝑊_∞,𝑝 (𝑝∈ℂ[𝑥]) is the central extension of the Lie algebra 𝒟𝑝(𝑡𝜕_𝑡) of differential operators on the circle that are a multiple of 𝑝(𝑡𝜕_𝑡).

This Lie algebra was studied by Kac and Liberati in [3]; observe that 𝑊_∞=𝑊_∞,𝑥. Following the ideas of Kac-Radul [2], in [3] they obtained the classification of the irreducible quasifinite highest weight modules over 𝑊_∞,𝑝. They also developed a general theory of quasifinite highest weight modules over ℤ-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 𝑊_∞ 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 𝑊_1+∞, 𝑊_∞,𝑝, and so forth.

The main goal of this work is to present a 𝑞-analog of Cheng-Wang [8] or a super-analog of Boyallian-Liberati [9]; namely, we classify the anti-involutions of the superalgebra 𝑞-𝒮𝒟 that preserve the gradation, where 𝑞-𝒮𝒟 is the superalgebra of regular pseudodifferential operators on the super circle 𝑆^1|1. 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 𝑊_1+∞, the matrix version, super analog and 𝑞-analog, and for the subalgebras constructed from the anti-involutions, showing the place of the present results in this long-term program.

Table 1 

The work is organized as follows. In Section 2, we classify the anti-involutions of 𝑞-𝒮𝒟 that preserve the gradation, where 𝑞-𝒮𝒟 is the superalgebra of regular pseudodifferential operators on the super circle 𝑆^1|1. 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 𝒮𝔖^as_𝑞 Preserving Its Principal Gradation

Let 𝑞∈ℂ^× with |𝑞|≠1. Now, 𝑇_𝑞 denotes the following operator on ℂ[𝑧⁻¹,𝑧]: 𝑇_𝑞𝑓(𝑧)=𝑓𝑞𝑧. (1) We denote by 𝔖^as_𝑞 the associative algebra of all pseudodifferential operators, that is, the operators on ℂ[𝑧⁻¹,𝑧] of the form 𝐸=_𝑘∈ℤ𝑒_𝑘(𝑧)𝑇^𝑘_𝑞,where𝑒_𝑖(𝑧)∈ℂ𝑧⁻¹,𝑧(sumisfinite). (2) Any pseudodifferential operator can be written as linear combinations of elements of the form 𝑧^𝑛𝑓(𝑇_𝑞), where 𝑓∈ℂ[𝑤⁻¹,𝑤] and 𝑛∈ℤ. The product in 𝔖^as_𝑞 is given by 𝑧^𝑛𝑓𝑇_𝑞⋅𝑧^𝑚𝑔𝑇_𝑞=𝑧^𝑛+𝑚𝑓𝑞^𝑚𝑇_𝑞𝑔𝑇_𝑞. (3) Letting 𝑤𝑡𝑧^𝑛𝑓(𝑇_𝑞)=𝑛, we define the principal ℤ-gradation of 𝔖^as_𝑞.

Moreover, we denote by 𝑀(1|1) the set of 2×2 supermatrices with coefficients in ℂ, viewed as the associative superalgebra of linear transformations of the complex (1|1)-dimensional superspace ℂ^(1|1). And we denote by 𝐸_𝑖𝑗 the 2×2 matrix with 1 in the 𝑖𝑗-place and 0 everywhere else. Declaring 𝐸₁₁, 𝐸₂₂ even and 𝐸₁₂, 𝐸₂₁ odd elements, we endow 𝑀(1|1) with a ℤ₂-gradation where |𝑀| denotes the parity of the homogeneous element 𝑀∈𝑀(1|1).

We denote by 𝒮𝔖^as_𝑞 the associative superalgebra of 2×2 supermatrices with entries in 𝔖^as_𝑞, namely, 𝒮𝔖^as_𝑞=𝔖^as_𝑞𝑀(1|1), (4) and the product is given by the usual matrix multiplication. Let 𝒮𝔖_𝑞 denote the Lie superalgebra obtained from 𝒮𝔖^as_𝑞 by taking the usual bracket.

Now, we introduce the linear map Str₀∶𝒮𝔖_𝑞→ℂ as Str_0𝑖𝑗𝑓_𝑖𝑗𝑇_𝑞𝐸_𝑖𝑗=𝑓₁₁𝑇_𝑞0−𝑓₂₂𝑇_𝑞0, (5) where (𝑓(𝑇_𝑞))₀=𝑓₀ if 𝑓(𝑇_𝑞)=∑_𝑘∈ℤ𝑓_𝑘𝑇^𝑘_𝑞, and we define the 2-cocycle 𝜓 in 𝒮𝔖_𝑞 by 𝜓𝑧^𝑛𝑓𝑇_𝑞𝐸_𝑖𝑗,𝑧^𝑚𝑔𝑇_𝑞𝐸_𝑘𝑙=−(−1)∑^𝑖𝑛−1_𝑟=0𝑓𝑞^−𝑛+𝑟𝑇_𝑞𝑔𝑞^𝑟𝑇_𝑞0𝛿_𝑗𝑘𝛿_𝑖𝑙,if𝑛=−𝑚>0,0,otherwise. (6) Then we denote by 𝒮𝔖_𝑞 the one-dimensional central extension of 𝒮𝔖_𝑞 with central charge 𝐶 corresponding to the 2-cocycle 𝜓, namely, 𝒮𝔖_𝑞=𝒮𝔖_𝑞⨁ℂ𝐶, where the bracket is given by 𝑧^𝑛𝑓𝑇_𝑞𝐸_𝑖𝑗,𝑧^𝑚𝑔𝑇_𝑞𝐸_𝑘𝑙=𝑧^𝑛+𝑚𝑓𝑞^𝑚𝑇_𝑞𝑔𝑇_𝑞𝛿_𝑗𝑘𝐸_𝑖𝑙−(−1)^{|𝐸𝑖𝑗||𝐸𝑘𝑙|}×𝑓𝑇_𝑞𝑔𝑞^𝑛𝑇_𝑞𝛿_𝑙𝑖𝐸_𝑘𝑗+𝜓𝑧^𝑛𝑓𝑇_𝑞𝐸_𝑖𝑗,𝑧^𝑚𝑔𝑇_𝑞𝐸_𝑘𝑙𝐶. (7)

Letting 𝑤𝑡(𝑧^𝑛𝑓(𝑇_𝑞)𝐸₁₂+𝑧^𝑛+1𝑔(𝑇_𝑞)𝐸₂₁)=𝑛+1/2,  𝑤𝑡𝑧^𝑛𝑓(𝑇_𝑞)𝐸_𝑖𝑖=𝑛  and 𝑤𝑡𝐶=0, where 𝑖=1,2,  𝑛∈ℤ, defines the principal (1/2)ℤ-gradation of 𝒮𝔖^as_𝑞, 𝒮𝔖_𝑞 and 𝒮𝔖_𝑞. This equips 𝒮𝔖^as_𝑞, 𝒮𝔖_𝑞, and 𝒮𝔖_𝑞 with (1/2)ℤ-gradations compatible with their ℤ₂-gradation; thus, 𝒮𝔖_𝑞=𝒮𝔖_𝑞0𝒮𝔖_𝑞1,𝒮𝔖_𝑞0=_𝑛∈ℤ𝒮𝔖_𝑞𝑛,𝒮𝔖_𝑞1=_𝑛∈ℤ𝒮𝔖_𝑞𝑛+1/2, (8) where 𝒮𝔖_𝑞𝑛=𝑧^𝑛𝑓₁𝑇_𝑞𝐸₁₁+𝑧^𝑛𝑓₂𝑇_𝑞𝐸₂₂∶𝑓_𝑖∈ℂ𝑤⁻¹,𝑤,𝑖=1,2+𝛿_𝑛,0ℂ𝐶,𝒮𝔖_𝑞𝑛+1/2=𝑧^𝑛𝑓₁𝑇_𝑞𝐸₁₂+𝑧^𝑛+1𝑓₂𝑇_𝑞𝐸₂₁∶𝑓_𝑖∈ℂ𝑤⁻¹,.,𝑤𝑖=1,2 (9)

An anti-involution 𝜎 of 𝒮𝔖^as_𝑞 is an involutive anti-automorphism of 𝒮𝔖^as_𝑞; that is, 𝜎∶𝒮𝔖^as_𝑞→𝒮𝔖^as_𝑞 with 𝜎²=𝐼𝑑, 𝜎(𝑏𝐴+𝐵)=𝑏𝜎(𝐴)+𝜎(𝐵)  and  𝜎(𝐴𝐵)=(−1)^|𝐴||𝐵|𝜎(𝐵)𝜎(𝐴), where 𝐴,𝐵∈𝒮𝔖^as_𝑞, and 𝑏∈ℂ.

The main result of this section is the following theorem with the classification of all anti-involutions of 𝒮𝔖^as_𝑞 that preserve the principal (1/2)ℤ-gradation.

Theorem 1. Any anti-involution 𝜎 of 𝒮𝔖^as_𝑞 which preserves the principal gradation is one of the following (𝑓∈ℂ[𝑤⁻¹,𝑤], 𝑛∈ℤ): (a)𝜎_{𝑎,𝑏,𝑐,𝑘}𝑧^𝑛𝑓𝑇_𝑞𝐸₁₁=𝑎^𝑛𝑞^{(𝑛−1)𝑛𝑘}𝑧^𝑛𝑓𝑏𝑞^−𝑛𝑇_𝑞⁻¹𝑇_𝑞^2𝑛𝑘𝐸₁₁,𝜎_{𝑎,𝑏,𝑐,𝑘}𝑧^𝑛𝑓𝑇_𝑞𝐸₂₂=𝑎^𝑛𝑞^{(𝑛−2)𝑛𝑘}𝑧^𝑛𝑓𝑏𝑞^−𝑛+1𝑇_𝑞⁻¹𝑇_𝑞^2𝑛𝑘𝐸₂₂,𝜎_{𝑎,𝑏,𝑐,𝑘}𝑧^𝑛𝑓𝑇_𝑞𝐸₁₂=𝑎^𝑛𝑐𝑞^𝑛2𝑘𝑧^𝑛+1𝑓𝑏𝑞^−𝑛𝑇_𝑞⁻¹𝑇_𝑞^{(2𝑛+1)𝑘}𝐸₂₁,𝜎_{𝑎,𝑏,𝑐,𝑘}𝑧^𝑛𝑓𝑇_𝑞𝐸₂₁=−𝑎^𝑛𝑐⁻¹𝑞^{((𝑛−1)2−𝑛)𝑘}𝑧^𝑛−1𝑓𝑏𝑞^−𝑛+1𝑇_𝑞⁻¹𝑇_𝑞^{(2𝑛−1)𝑘}𝐸₁₂, (10) with 𝑎,𝑏,𝑐∈ℂ^×, 𝑘∈ℤ such that −𝑎𝑏^𝑘=𝑞^𝑘; (b)𝜎_{𝑎,𝑏,𝑐,𝑘,𝑙}𝑧^𝑛𝑓𝑇_𝑞𝐸₁₁=𝑎^𝑛𝑞^{(𝑛(𝑛−1)/2)𝑘}𝑧^𝑛𝑓𝑏𝑞^−𝑛𝑇_𝑞⁻¹𝑇_𝑞^𝑛𝑘𝐸₂₂,𝜎_{𝑎,𝑏,𝑐,𝑘,𝑙}𝑧^𝑛𝑓𝑇_𝑞𝐸₂₂=𝑎^𝑛𝑞^{(𝑛(𝑛−1)/2)𝑘+𝑛𝑙}𝑧^𝑛𝑓𝑏𝑞^−𝑛𝑇_𝑞⁻¹𝑇_𝑞^𝑛𝑘𝐸₁₁,𝜎_{𝑎,𝑏,𝑐,𝑘,𝑙}𝑧^𝑛𝑓𝑇_𝑞𝐸₁₂=𝑎^𝑛𝑐𝑞^{(𝑛(𝑛−1)/2)𝑘+𝑛𝑙}𝑧^𝑛𝑓𝑏𝑞^−𝑛𝑇_𝑞⁻¹𝑇_𝑞^{𝑛𝑘+𝑙}𝐸₁₂,𝜎_{𝑎,𝑏,𝑐,𝑘,𝑙}𝑧^𝑛𝑓𝑇_𝑞𝐸₂₁=−𝑎^𝑛𝑐⁻¹𝑞^{(𝑛(𝑛−1)/2)𝑘}𝑧^𝑛𝑓𝑏𝑞^−𝑛𝑇_𝑞⁻¹𝑇_𝑞^{𝑛𝑘−𝑙}𝐸₂₁, (11) with 𝑎,𝑏,𝑐∈ℂ^×, 𝑘,𝑙∈ℤ such that 𝑎²𝑏^𝑘=𝑞^𝑘−𝑙 and 𝑏^𝑙𝑐²=1.

We divide the proof of Theorem 1 into several results.

Let 𝜎 be an anti-involution of 𝒮𝔖^as_𝑞 which preserves the principal gradation; then, 𝜎 defines linear maps 𝜎_𝑖,𝑗∶𝔖^as_𝑞→𝔖^as_𝑞 preserving the principal gradation 𝜎𝑧^𝑛𝑓𝑇_𝑞𝐸_𝑖𝑖=𝜎_𝑖,1𝑧^𝑛𝑓𝑇_𝑞𝐸₁₁+𝜎_𝑖,2𝑧^𝑛𝑓𝑇_𝑞𝐸₂₂. (12)

Lemma 2. Let 𝜎 be an anti-involution of 𝒮𝔖_𝑞^𝑎𝑠 preserving the principal gradation. Then 𝜎 satisfies one of the following conditions: (a)𝜎𝑧^𝑛𝑓𝑇_𝑞𝐸₁₁=𝜎_1,1𝑧^𝑛𝑓𝑇_𝑞𝐸₁₁,𝜎𝑧^𝑛𝑓𝑇_𝑞𝐸₂₂=𝜎_2,2𝑧^𝑛𝑓𝑇_𝑞𝐸₂₂,𝜎𝐸₁₂=𝑧𝑓₂₁𝑇_𝑞𝐸₂₁,𝜎𝐸₂₁=𝑧⁻¹𝑔₁₂𝑇_𝑞𝐸₁₂; (13)(b)𝜎𝑧^𝑛𝑓𝑇_𝑞𝐸₁₁=𝜎_1,2𝑧^𝑛𝑓𝑇_𝑞𝐸₂₂,𝜎𝑧^𝑛𝑓𝑇_𝑞𝐸₂₂=𝜎_2,1𝑧^𝑛𝑓𝑇_𝑞𝐸₁₁,𝜎𝐸₁₂=𝑓₁₂𝑇_𝑞𝐸₁₂,𝜎𝐸₂₁=𝑔₂₁𝑇_𝑞𝐸₂₁. (14) For some 𝑓_𝑖𝑗,𝑔_𝑖𝑗∈ℂ[𝑤⁻¹,𝑤], where 1⩽𝑖, 𝑗⩽2 and 𝑖≠𝑗.

Proof. Since 𝜎 preserves the principal gradation, we have that 𝜎𝐸₁₂=𝑓₁₂𝑇_𝑞𝐸₁₂+𝑧𝑓₂₁𝑇_𝑞𝐸₂₁,𝜎𝐸₂₁=𝑧⁻¹𝑔₁₂𝑇_𝑞𝐸₁₂+𝑔₂₁𝑇_𝑞𝐸₂₁, (15) for some 𝑓_𝑖𝑗,𝑔_𝑖𝑗∈ℂ[𝑤⁻¹,𝑤], with 1⩽𝑖, 𝑗⩽2 and 𝑖≠𝑗.
Then, since 0=𝜎(𝑧^𝑛𝑓(𝑇_𝑞)𝐸₁₁⋅𝑧^𝑚𝑔(𝑇_𝑞)𝐸₂₂)=𝜎(𝑧^𝑚𝑔(𝑇_𝑞)𝐸₂₂)𝜎(𝑧^𝑛𝑓(𝑇_𝑞)𝐸₁₁), we have two possibilities 𝜎𝑧^𝑛𝑓𝑇_𝑞𝐸₁₁=𝜎_1,1𝑧^𝑛𝑓𝑇_𝑞𝐸₁₁,𝜎𝑧^𝑛𝑓𝑇_𝑞𝐸₂₂=𝜎_2,2𝑧^𝑛𝑓𝑇_𝑞𝐸₂₂, (16) or 𝜎𝑧^𝑛𝑓𝑇_𝑞𝐸₁₁=𝜎_1,2𝑧^𝑛𝑓𝑇_𝑞𝐸₂₂,𝜎𝑧^𝑛𝑓𝑇_𝑞𝐸₂₂=𝜎_2,1𝑧^𝑛𝑓𝑇_𝑞𝐸₁₁. (17) Moreover, since 0=𝜎(𝐸²_𝑖𝑗)=−𝜎(𝐸_𝑖𝑗)², for 1⩽𝑖,𝑗⩽2, 𝑖≠𝑗, from (15) we have that 𝜎𝐸₁₂=𝑓₁₂𝑇_𝑞𝐸₁₂or𝜎𝐸₁₂=𝑧𝑓₂₁𝑇_𝑞𝐸₂₁,𝜎𝐸₂₁=𝑔₂₁𝑇_𝑞𝐸₂₁or𝜎𝐸₂₁=𝑧⁻¹𝑔₁₂𝑇_𝑞𝐸₁₂. (18) Finally, since 𝜎𝐸₁₁=−𝜎𝐸₂₁𝜎𝐸₁₂, (19) the result follows from (16), (17), (18), and (19).

Corollary 3. Let 𝜎 be an anti-involution of 𝒮𝔖^as_𝑞 which preserves the principal gradation. Then one has the following. (a)If 𝜎 satisfies Lemma 2(a), then 𝜎_𝑖,𝑗(1)=𝛿_𝑖,𝑗, for all 1⩽𝑖, 𝑗⩽2. (b)If 𝜎 satisfies Lemma 2(b), then 𝜎_𝑖,𝑗(1)=1−𝛿_𝑖,𝑗, for all 1⩽𝑖, 𝑗⩽2.

Proof. It is clear that 𝜎(𝐼𝑑)=𝐼𝑑; then the result follows from Lemma 2.

Proposition 4. Let 𝜎 be an anti-involution of 𝒮𝔖_𝑞^𝑎𝑠 which preserves the principal gradation and let one assume that it satisfies Lemma 2(a). Then 𝜎 is one of the 𝜎_{𝑎,𝑏,𝑐,𝑘} from Theorem 1(a).

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

Lemma 5. Let 𝜎 be as in Proposition 4; then, 𝜎𝑧^𝑛𝑓𝑇_𝑞𝐸_𝑖𝑖=𝑎^𝑛_𝑖𝑞^{(𝑛(𝑛−1)/2)𝑘𝑖}𝑧^𝑛𝑓𝑏_𝑖𝑞^−𝑛𝑇_𝑞⁻¹𝑇^𝑛𝑘𝑖_𝑞𝐸_𝑖𝑖,𝜎𝑧^𝑛𝑓𝑇_𝑞𝐸₁₂=𝑎^𝑛₁𝑐𝑞^{(𝑛(𝑛−1)/2)𝑘1+𝑛𝑘}𝑧^𝑛+1𝑓𝑏₁𝑞^−𝑛𝑇_𝑞⁻¹𝑇^𝑛𝑘1_𝑞^+𝑘𝐸₂₁,𝜎𝑧^𝑛𝑓𝑇_𝑞𝐸₂₁=−𝑎₂𝑐⁻¹𝑞^{(𝑛𝑘2/2−𝑘)(𝑛−1)}𝑧^𝑛−1𝑓𝑏₂𝑞^−𝑛𝑇_𝑞⁻¹𝑇^𝑛𝑘2_𝑞^−𝑘𝐸₁₂, (20) where 𝑎_𝑖,𝑏_𝑖,𝑐∈ℂ^×, 𝑘_𝑖,𝑘∈ℤ such that 𝑎²_𝑖𝑏^𝑘𝑖_𝑖=𝑞^𝑘𝑖, with 𝑖=1,2.

Proof. It is easy to check that 𝜎_𝑖,𝑖 is a anti-involution of 𝔖^as_𝑞 with 𝑖=1,2; then, from Section 3 in [9] and hypothesis, we obtain that 𝜎𝑧^𝑛𝑓𝑇_𝑞𝐸_𝑖𝑖=𝑎^𝑛_𝑖𝑞^{(𝑛(𝑛−1)/2)𝑘𝑖}𝑧^𝑛𝑓𝑏_𝑖𝑞^−𝑛𝑇_𝑞⁻¹𝑇^𝑛𝑘𝑖_𝑞𝐸_𝑖𝑖, (21) with 𝑘_𝑖∈ℤ, 𝑎_𝑖,𝑏_𝑖∈ℂ^× such that 𝑎²_𝑖𝑏^𝑘𝑖_𝑖=𝑞^𝑘𝑖, 𝑖=1,2. Besides, from Corollary 3(a) and hypothesis, we have that 𝐸₁₁=−𝜎𝐸₂₁𝜎𝐸₁₂=−𝑔₁₂𝑞𝑇_𝑞𝑓₂₁𝑇_𝑞𝐸₁₁. (22) Therefore 𝑓₂₁(𝑇_𝑞)=𝑐𝑇^𝑘_𝑞 and 𝑔₁₂(𝑇_𝑞)=−𝑐⁻¹𝑞^𝑘𝑇_𝑞^−𝑘 with 𝑐∈ℂ^×, 𝑘∈ℤ; then, 𝜎𝐸₁₂=𝑐𝑧𝑇^𝑘_𝑞𝐸₂₁,𝜎𝐸₂₁=−𝑐⁻¹𝑞^𝑘𝑧⁻¹𝑇_𝑞^−𝑘𝐸₁₂. (23) Moreover, 𝜎𝑧^𝑛𝑓𝑇_𝑞𝐸_𝑖𝑗=𝜎𝐸_𝑖𝑗𝜎𝑧^𝑛𝑓𝑇_𝑞𝐸_𝑖𝑖, (24) with 𝑖,𝑗=1,2, 𝑖≠𝑗. The proof follows from (21) and by replacing (21) and (23) in (24).

Proof of Proposition 4. From Lemma 5, we have that 𝐸₁₂=𝜎²𝐸₁₂=𝑐𝜎𝐸₂₁𝜎𝑧𝑇^𝑘_𝑞𝐸₂₂=−𝑎₂𝑏^𝑘₂𝑞^−𝑘𝑇^𝑘2_𝑞^−2𝑘𝐸₁₂,𝐸₁₂=𝜎²𝐸₁₂=𝑐𝜎𝑧𝑇^𝑘_𝑞𝐸₁₁𝜎𝐸₂₁=−𝑎₁𝑏^𝑘₁𝑞^𝑘−𝑘1𝑇^𝑘1_𝑞^−2𝑘𝐸₁₂. (25)(26) Then from (25)-(26), we obtain that 𝑘₁=𝑘₂=2𝑘, (27) and also from (26)-(27) we obtain −𝑎₁𝑏^𝑘₁=𝑞^𝑘. (28) Then, from (27) and again using Lemma 5, we have that 𝜎𝑧^𝑛𝑇^𝑙_𝑞𝐸₁₂=𝑎^𝑛₁𝑏^𝑙₁𝑐𝑞^{(𝑛𝑘−𝑙)𝑛}𝑧^𝑛+1𝑇_𝑞^{(2𝑛+1)𝑘−𝑙}𝐸₂₁,𝜎𝑧^𝑛𝑇^𝑙_𝑞𝐸₁₂=𝜎𝑧^𝑛𝑇^𝑙_𝑞𝐸₂₂𝜎𝐸₁₂=𝑎^𝑛₂𝑏^𝑙₂𝑐𝑞^{(𝑛𝑘−𝑙)(𝑛+1)}𝑧^𝑛+1𝑇_𝑞^{(2𝑛+1)𝑘−𝑙}𝐸₂₁, (29)(30) for all 𝑛,𝑙∈ℤ. Comparing (29) with (30), we obtain that 𝑎^𝑛₁𝑏^𝑙₁=𝑎^𝑛₂𝑏^𝑙₂𝑞^{𝑛𝑘−𝑙},∀𝑛,𝑙∈ℤ; (31) in particular, if𝑛=0,𝑙=1,𝑏₂=𝑏₁𝑞,if𝑛=1,𝑙=0,𝑎₂=𝑎₁𝑞^−𝑘. (32)
From (28) and replacing (27) and (32) in (20), we obtain that 𝜎=𝜎_{𝑎1,𝑏1,𝑐,𝑘}, finishing the proof.

Proposition 6. Let 𝜎 be an anti-involution of 𝒮𝔖_𝑞^𝑎𝑠 which preserves the principal graduation, and let one assume that it satisfies Lemma 2(b). Then 𝜎 is one of the 𝜎_{𝑎,𝑏,𝑐,𝑘,𝑙} from Theorem 1(b).

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

Lemma 7. Let 𝜎 be as in Proposition 6, then 𝜎𝑧^𝑛𝑓𝑇_𝑞𝐸₁₁=𝑎^𝑛₁𝑞^{(𝑛(𝑛−1)/2)𝑘1}𝑧^𝑛𝑓𝑏₁𝑞^𝑛𝑚1𝑇^𝑚1_𝑞𝑇^𝑛𝑘1_𝑞𝐸₂₂,𝜎𝑧^𝑛𝑓𝑇_𝑞𝐸₂₂=𝑎^𝑛₂𝑞^{(𝑛(𝑛−1)/2)𝑘2}𝑧^𝑛𝑓𝑏₂𝑞^𝑛𝑚2𝑇^𝑚2_𝑞𝑇^𝑛𝑘2_𝑞𝐸₁₁,𝜎𝑧^𝑛𝑓𝑇_𝑞𝐸₁₂=𝑎^𝑛₁𝑐𝑞^{(𝑛(𝑛−1)/2)𝑘1+𝑛𝑙}𝑧^𝑛𝑓𝑏₁𝑞^𝑛𝑚1𝑇^𝑚1_𝑞𝑇^𝑛𝑘1_𝑞^+𝑙𝐸₁₂,𝜎𝑧^𝑛𝑓𝑇_𝑞𝐸₂₁=−𝑎₂𝑐⁻¹𝑞^{(𝑛(𝑛−1)/2)𝑘2−𝑛𝑙}𝑧^𝑛𝑓𝑏₂𝑞^𝑛𝑚2𝑇^𝑚2_𝑞𝑇^𝑛𝑘2_𝑞^−𝑙𝐸₂₁, (33) where 𝑎_𝑖,𝑏_𝑖,𝑐∈ℂ^×, 𝑘_𝑖,𝑚_𝑖,𝑙∈ℤ, and 𝑖=1,2.

Proof. From hypothesis and Corollary 3(b), we have that 𝐸₁₁=𝜎𝑇_𝑞𝐸₂₂𝑇_𝑞⁻¹𝐸₂₂=𝜎_2,1𝑇_𝑞⁻¹𝜎_2,1𝑇_𝑞𝐸₁₁,𝐸₂₂=𝜎𝑇_𝑞𝐸₁₁𝑇_𝑞⁻¹𝐸₁₁=𝜎_1,2𝑇_𝑞⁻¹𝜎_1,2𝑇_𝑞𝐸₂₂,𝐸₁₁=𝜎𝑧𝐸₂₂𝑧⁻¹𝐸₂₂=𝜎_2,1𝑧⁻¹𝜎_2,1(𝑧)𝐸₁₁,𝐸₂₂=𝜎𝑧𝐸₁₁𝑧⁻¹𝐸₁₁=𝜎_1,2𝑧⁻¹𝜎_1,2(𝑧)𝐸₂₂,𝐸₁₁=−𝜎𝐸₁₂𝜎𝐸₂₁=−𝑓₁₂𝑇_𝑞𝑔₂₁𝑇_𝑞𝐸₁₁. (34)(35) Then, using (34), we have that 𝜎_𝑖,𝑗𝑇_𝑞^±1=𝑏_𝑖^±1𝑇^±𝑚𝑖_𝑞,𝜎_𝑖,𝑗(𝑧)=𝑎_𝑖𝑧𝑇^𝑘𝑖_𝑞,𝜎_𝑖,𝑗𝑧⁻¹=𝑎_𝑖⁻¹𝑞^𝑘𝑖𝑧⁻¹𝑇^−𝑘𝑖_𝑞, (36) and by induction we obtain that 𝜎𝑇^𝑚_𝑞𝐸_𝑖𝑖=𝑏^𝑚_𝑖𝑇^𝑚𝑖𝑚_𝑞𝐸_𝑗𝑗𝜎𝑧,∀𝑚∈ℤ,_𝑛𝐸_𝑖𝑖=𝑎^𝑛_𝑖𝑞^{(𝑛(𝑛−1)/2)𝑘𝑖}𝑧^𝑛𝑇^𝑛𝑘𝑖_𝑞𝐸_𝑗𝑗,∀𝑛∈ℤ, (37) where 𝑎_𝑖,𝑏_𝑖∈ℂ^×, 𝑘_𝑖,𝑚_𝑖∈ℤ, 𝑖,𝑗=1,2,  𝑖≠𝑗. Then, from (37) we have that 𝜎𝑧^𝑛𝑓𝑇_𝑞𝐸_𝑖𝑖=𝜎𝑓𝑇_𝑞𝐸_𝑖𝑖𝜎𝑧^𝑛𝐸_𝑖𝑖=𝑎^𝑛_𝑖𝑞^{(𝑛(𝑛−1)/2)𝑘𝑖}𝑧^𝑛𝑓𝑏_𝑖𝑞^𝑛𝑚𝑖𝑇^𝑚𝑖_𝑞𝑇^𝑛𝑘𝑖_𝑞𝐸_𝑗𝑗, (38) with 𝑖,𝑗=1,2, 𝑖≠𝑗.
On the other hand, from (35) we have that 𝑓₁₂(𝑇_𝑞)=𝑐𝑇^𝑙_𝑞 and 𝑔₂₁(𝑇_𝑞)=−𝑐⁻¹𝑇_𝑞^−𝑙 with 𝑐∈ℂ^× and 𝑙∈ℤ; therefore, 𝜎𝐸₁₂=𝑐𝑇^𝑙_𝑞𝐸₁₂,𝜎𝐸₂₁=−𝑐⁻¹𝑇_𝑞^−𝑙𝐸₂₁. (39) Moreover, 𝜎𝑧^𝑛𝑓𝑇_𝑞𝐸_𝑖𝑗=𝜎𝐸_𝑖𝑗𝜎𝑧^𝑛𝑓𝑇_𝑞𝐸_𝑖𝑖, (40) with 𝑖,𝑗=1,2,𝑖≠𝑗. The proof follows from (38) and by replacing (38) and (39) in (40).

Proof of Proposition 6. By Lemma 7, we have that 𝑇_𝑞𝐸₁₁=𝜎²𝑇_𝑞𝐸₁₁=𝑏₁𝑏^𝑚1₂𝑇^𝑚1𝑚2_𝑞𝐸₁₁,𝑧𝐸_𝑖𝑖=𝜎²𝑧𝐸_𝑖𝑖=𝑎_𝑖𝑎_𝑗𝑏^𝑘𝑖_𝑗𝑞^{𝑚𝑗𝑘𝑖}𝑧𝑇^{𝑚𝑗𝑘𝑖+𝑘𝑗}_𝑞𝐸_𝑖𝑖,𝐸₁₂=𝜎²𝐸₁₂=𝑏^𝑙₁𝑐²𝐸₁₂,𝜎𝑧𝐸₁₂=𝑎₁𝑐𝑞^𝑙𝑧𝑇^𝑘1_𝑞^+𝑙𝐸₁₂,𝜎𝑧𝐸₁₂=𝜎𝑧𝐸₂₂𝜎𝐸₁₂=𝑎₂𝑐𝑧𝑇^𝑘1_𝑞^+𝑙𝐸₁₂. (41)(42)(43) Then, using (41), we have that 𝑚₁=𝑚₂=1or𝑚₁=𝑚₂𝑏=−1,₁=𝑏^−𝑚1₂,𝑎_𝑖𝑎_𝑗𝑏^𝑘𝑖_𝑗𝑞^{𝑚𝑗𝑘𝑖}𝑚=1,_𝑗𝑘_𝑖=−𝑘_𝑗,𝑏^𝑙₁𝑐²=1, (44)(45)(46)(47)(48) where 𝑖,𝑗=1,2, 𝑖≠𝑗. Suppose that 𝑚₁=𝑚₂=1; then, from (45) and (47) we have that 𝑏₁=𝑏₂⁻¹, 𝑘₁=−𝑘₂, and by replacing them in (46), we obtain that 𝑞^2𝑘1=1, but since 𝑞 is not a root of unity, we necessarily have that 𝑘₁=𝑘₂=0. Then, using Lemma 7 we get 𝜎𝑧𝑇_𝑞𝐸₁₁=𝑎₁𝑏₁𝑞𝑧𝑇_𝑞𝐸₂₂; (49) moreover, 𝜎𝑧𝑇_𝑞𝐸₁₁=𝑞⁻¹𝜎𝑧𝐸₁₁𝜎𝑇_𝑞𝐸₁₁=𝑎₁𝑏₁𝑞⁻¹𝑧𝑇_𝑞𝐸₂₂. (50) Comparing (49) with (50), we obtain that 𝑞²=1 which is a contradiction. Therefore, by (44) and the aforementioned result, we get that 𝑚₁=𝑚₂=−1. (51) Then replacing (51) in (45) and (47) we obtain that 𝑏₁=𝑏₂,𝑘₁=𝑘₂. (52) On the other hand, comparing (42) with (43), we have that 𝑎₂=𝑎₁𝑞^𝑙, (53) and by replacing (51), (52), and (53) in (46), we obtain that 𝑎²₁𝑏^𝑘1₁=𝑞^𝑘1−𝑙. (54) Using (48) and (54) and by replacing (51), (52), and (53) in (33), we obtain that 𝜎=𝜎_{𝑎1,𝑏1,𝑐,𝑘1,𝑙}, 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 𝒮𝔖^as_𝑞 which preserves the principal gradation satisfies (a) or (b), finishing the proof.

Now, given an anti-involution 𝜎 of 𝒮𝔖^as_𝑞, one can check that the set of points minus 𝜎-fixed is a subalgebra of 𝒮𝔖_𝑞. Moreover, if 𝜎 preserves the principal (1/2)ℤ-graduation, this subalgebra inherits the (1/2)ℤ-graduation. These subalgebras are described in the last part of this section.

We define the following automorphisms of 𝒮𝔖^as_𝑞 by Θ_𝑠(𝑀)=𝑧^−𝑠𝐼𝑑⋅𝑀⋅𝑧^𝑠𝐼𝑑,∀𝑀∈𝒮𝔖^as_𝑞,Φ_{𝑠(𝒮𝔖as𝑞)0}Φ=𝑖𝑑,_𝑠𝑧^𝑛𝑓𝑇_𝑞𝐸₁₂+𝑧^𝑛+1𝑔𝑇_𝑞𝐸₂₁=𝑠𝑧^𝑛𝑓𝑇_𝑞𝐸₁₂+𝑠⁻¹𝑧^𝑛+1𝑔𝑇_𝑞𝐸₂₁, (55) with 𝑠∈ℂ^×. On the other hand, given 𝑛,𝑚∈ℤ, we denote ℂ𝑤⁻¹,𝑤^(𝑛,𝑚)∶=𝑓∈ℂ𝑤⁻¹,𝑤∶𝑓(𝑤)=−(−1)^𝑛𝑓𝑤⁻¹𝑤^𝑚. (56)

Remark 8. We see that 𝑓(𝑤)∈ℂ[𝑤⁻¹,𝑤]^(𝑛,𝑚) with 𝑓(𝑤)=∑𝑓_𝑗𝑤^𝑗 if and only if 𝑓_𝑗=−(−1)^𝑛𝑓_𝑚−𝑗, for all 𝑗∈ℤ.

The Case 𝜎_{𝑎,𝑏,𝑐,𝑘}. Let 𝑎,𝑏,𝑐∈ℂ^×, 𝑘∈ℤ, be such that −𝑎𝑏^𝑘=𝑞^𝑘. We denote by 𝒮𝔖^{𝑎,𝑏,𝑐}_𝑞,𝑘 the Lie subalgebra of 𝒮𝔖_𝑞 consisting of −𝜎_{𝑎,𝑏,𝑐,𝑘}-fixed points; then, it inherits a (1/2)ℤ-gradation from 𝒮𝔖_𝑞; therefore, 𝒮𝔖^{𝑎,𝑏,𝑐}_𝑞,𝑘=⨁_{𝑗∈(1/2)ℤ}(𝒮𝔖^{𝑎,𝑏,𝑐}_𝑞,𝑘)_𝑗, where 𝒮𝔖^{𝑎,𝑏,𝑐}_{𝑞,𝑘𝑗}=𝑀∈𝒮𝔖_𝑞𝑗∶𝜎_{𝑎,𝑏,𝑐,𝑘}(𝑀)=−𝑀. (57) Moreover, we denote 𝒮𝔖_𝑞,𝑘∶=𝒮𝔖^{−𝑞𝑘,1,1}_𝑞,𝑘 and 𝜎_𝑞,𝑘∶=𝜎_{−𝑞𝑘,1,1,𝑘}.

The following lemma gives a description of 𝒮𝔖^{𝑎,𝑏,𝑐}_𝑞,𝑚.

Lemma 9. Let 𝑎, 𝑏, 𝑐, 𝑘 be as aforementioned; then, 𝒮𝔖^{𝑎,𝑏,𝑐}_𝑞,𝑘≃𝒮𝔖_𝑞,𝑘 and 𝒮𝔖_{𝑞,𝑘𝑛}=𝑧^𝑛𝑓𝑞^𝑛/2𝑇_𝑞𝐸₁₁+𝑧^𝑛𝑔𝑞^(𝑛−1)/2𝑇_𝑞𝐸₂₂∶𝑓,𝑔∈ℂ𝑤⁻¹,𝑤^{(𝑛,2𝑛𝑘)},𝒮𝔖_{𝑞,𝑘𝑛+1/2}=𝑧^𝑛𝑓𝑇_𝑞𝐸₁₂−(−1)^𝑛𝑞^{(𝑛+1)𝑛𝑘}𝑧^𝑛+1𝑓𝑞^−𝑛𝑇_𝑞⁻¹𝑇_𝑞^{(2𝑛+1)𝑘}𝐸₂₁∶𝑓∈ℂ𝑤⁻¹,𝑤, (58) for all 𝑛∈ℤ.

Proof. First, it is easy to check that Θ_−𝑠𝜎_{𝑎,𝑏,𝑐,𝑘}Θ_𝑠=𝜎_{𝑎𝑞−2𝑘𝑠,𝑏𝑞2𝑠,𝑐𝑞−𝑘𝑠,𝑘},Φ_−𝑡𝜎_{𝑎,𝑏,𝑐,𝑘}Φ_𝑡=𝜎_{𝑎,𝑏,𝑐𝑡2,𝑘}. (59) Then if we take 𝑠,𝑡∈ℂ^× such that 𝑏⁻¹=𝑞^2𝑠 and 𝑡²=𝑐⁻¹𝑞^𝑘𝑠, from (59) and the relation between 𝑎,𝑏,𝑐,𝑘, we obtain that 𝒮𝔖^{𝑎,𝑏,𝑐}_𝑞,𝑘≃𝒮𝔖_𝑞,𝑘.
On the other hand, by Theorem 1(a) and linearity of 𝜎_𝑞,𝑘 we obtain that (for 𝑛∈ℤ) 𝑧^𝑛𝑓(𝑇_𝑞)𝐸₁₁+𝑧^𝑛𝑔(𝑇_𝑞)𝐸₂₂∈(𝒮𝔖_𝑞,𝑘)_𝑛 if and only if 𝑧^𝑛𝑓(𝑇_𝑞)𝐸₁₁, 𝑧^𝑛𝑔(𝑇_𝑞)𝐸₂₂∈(𝒮𝔖_𝑞,𝑘)_𝑛. Now, let 𝑧^𝑛ℎ(𝑇_𝑞)𝐸₁₁∈(𝒮𝔖_𝑞,𝑘)_𝑛 be with ℎ(𝑤)=∑ℎ_𝑗𝑤^𝑗; then, −𝑧^𝑛ℎ𝑇_𝑞𝐸₁₁=𝜎_𝑞,𝑘𝑧^𝑛ℎ𝑇_𝑞𝐸₁₁=(−1)^𝑛𝑞^𝑛2𝑘𝑧^𝑛ℎ𝑞^−𝑛𝑇_𝑞⁻¹𝑇_𝑞^2𝑛𝑘𝐸₁₁ (60) if and only if 𝑞^{−(𝑛/2)𝑗}ℎ_𝑗=−(−1)^𝑛𝑞^{(−𝑛/2)(2𝑛𝑘−𝑗)}ℎ_{2𝑛𝑘−𝑗}, for all 𝑗∈ℤ, which is equivalent to 𝑓(𝑤)=ℎ(𝑞^−𝑛/2𝑤)∈ℂ[𝑤⁻¹,𝑤]^{(𝑛,2𝑛𝑘)} (see Remark 8). Therefore, 𝑧^𝑛ℎ𝑇_𝑞𝐸₁₁=𝑧^𝑛𝑓𝑞^𝑛/2𝑇_𝑞𝐸₁₁,with𝑓(𝑤)∈ℂ𝑤⁻¹,𝑤^{(𝑛,2𝑛𝑘)}. (61) Similarly, we prove that 𝑧^𝑛ℎ(𝑇_𝑞)𝐸₂₂∈(𝒮𝔖_𝑞,𝑘)_𝑛 if and only if 𝑧^𝑛ℎ𝑇_𝑞𝐸₂₂=𝑧^𝑛𝑔𝑞^(𝑛−1)/2𝑇_𝑞𝐸₂₂,where𝑔(𝑤)∈ℂ𝑤⁻¹,𝑤^{(𝑛,2𝑛𝑘)}. (62) Now, we suppose that 𝑧^𝑛𝑓(𝑇_𝑞)𝐸₁₂+𝑧^𝑛+1𝑔(𝑇_𝑞)𝐸₂₁∈(𝒮𝔖_𝑞,𝑘)_𝑛+1/2; then, −𝑧^𝑛𝑓𝑇_𝑞𝐸₁₂−𝑧^𝑛+1𝑔𝑇_𝑞𝐸₂₁=𝜎_𝑞,𝑘𝑧^𝑛𝑓𝑇_𝑞𝐸₁₂+𝑧^𝑛+1𝑔𝑇_𝑞𝐸₂₁=(−1)^𝑛𝑞^𝑛2𝑘𝑧^𝑛𝑔𝑞^−𝑛𝑇_𝑞⁻¹𝑇_𝑞^{(2𝑛+1)𝑘}𝐸₁₂+𝑞^𝑛𝑘𝑧^𝑛+1𝑓𝑞^−𝑛𝑇_𝑞⁻¹𝑇_𝑞^{(2𝑛+1)𝑘}𝐸₂₁, (63) if and only if 𝑔(𝑇_𝑞)=−(−1)^𝑛𝑞^{(𝑛+1)𝑛𝑘}𝑓(𝑞^−𝑛𝑇_𝑞⁻¹)𝑇_𝑞^{(2𝑛+1)𝑘}, finishing the proof.

The Case 𝜎_{𝑎,𝑏,𝑐,𝑘,𝑙}. Let 𝑎,𝑏,𝑐∈ℂ^×, 𝑘,𝑙∈ℤ, be such that 𝑎²𝑏^𝑘=𝑞^𝑘−𝑙 and 𝑏^𝑙𝑐²=1. We denote by 𝒮𝔖^{𝑎,𝑏,𝑐}_{𝑞,𝑘,𝑙} the Lie subalgebra of 𝒮𝔖_𝑞 consisting of −𝜎_{𝑎,𝑏,𝑐,𝑘,𝑙}-fixed; then, it inherits a (1/2)ℤ-gradation from 𝒮𝔖_𝑞; thus 𝒮𝔖^{𝑎,𝑏,𝑐}_{𝑞,𝑘,𝑙}=⨁_{𝑗∈(1/2)ℤ}(𝒮𝔖^{𝑎,𝑏,𝑐}_{𝑞,𝑘,𝑙})_𝑗, where 𝒮𝔖^{𝑎,𝑏,𝑐}_{𝑞,𝑘,𝑙𝑗}=𝑀∈𝒮𝔖_𝑞𝑗∶𝜎_{𝑎,𝑏,𝑐,𝑘,𝑙}(𝑀)=−𝑀. (64) Moreover, we denote 𝒮𝔖^+±_{𝑞,𝑘,𝑙}∶=𝒮𝔖^𝑎,1,±1_{𝑞,𝑘,𝑙}, 𝜎^+±_{𝑞,𝑘,𝑙}∶=𝜎_{𝑎,1,±1,𝑘,𝑙} and 𝒮𝔖^−,±_{𝑞,𝑘,𝑙}∶=𝒮𝔖^{−𝑎,1,±1}_{𝑞,𝑘,𝑙}, 𝜎^−,±_{𝑞,𝑘,𝑙}∶=𝜎_{−𝑎,1,±1,𝑘,𝑙}, with 𝑎=𝑞^{(𝑘−𝑙)/2}. The following lemma gives a description of 𝒮𝔖^{𝑎,𝑏,𝑐}_{𝑞,𝑘,𝑙}. We will need the following notation: 𝛿_𝑛∶=0,in𝒮𝔖^+,+_{𝑞,𝑘,𝑙𝑛+1/2};1,in𝒮𝔖^+,−_{𝑞,𝑘,𝑙𝑛+1/2};𝑛,in𝒮𝔖^−,+_{𝑞,𝑘,𝑙𝑛+1/2};𝑛+1,in𝒮𝔖^−,−_{𝑞,𝑘,𝑙𝑛+1/2}. (65)

Lemma 10. Let 𝑎, 𝑏, 𝑐, 𝑘, 𝑙 be as aforementioned; then, 𝒮𝔖^{𝑎,𝑏,𝑐}_{𝑞,𝑘,𝑙} is isomorphic to some of the following algebras: 𝒮𝔖^+,+_{𝑞,𝑘,𝑙}, 𝒮𝔖^+,−_{𝑞,𝑘,𝑙}, 𝒮𝔖^−,+_{𝑞,𝑘,𝑙}, or 𝒮𝔖^−,−_{𝑞,𝑘,𝑙}. And 𝒮𝔖^+±_{𝑞,𝑘,𝑙𝑛}=𝑧^𝑛𝑓𝑇_𝑞𝐸₁₁−(±1)^𝑛𝑞^{𝑛(𝑛𝑘−𝑙)/2}×𝑧^𝑛𝑓𝑞^−𝑛𝑇_𝑞⁻¹𝑇_𝑞^𝑛𝑘𝐸₂₂∶𝑓∈ℂ𝑤⁻¹,𝑤,𝒮𝔖^+±_{𝑞,𝑘,𝑙𝑛+1/2}=𝑧^𝑛𝑓𝑞^𝑛/2𝑇_𝑞𝐸₁₂+𝑧^𝑛+1𝑔𝑞^(𝑛+1)/2𝑇_𝑞×𝐸₂₁∶𝑓∈ℂ𝑤⁻¹,𝑤^{(𝛿𝑛,𝑛𝑘+𝑙)},𝑔∈ℂ𝑤⁻¹,𝑤^{(𝛿𝑛+1,(𝑛+1)𝑘−𝑙)},𝒮𝔖^−±_{𝑞,𝑘,𝑙𝑛}=𝑧^𝑛𝑓𝑇_𝑞𝐸₁₁−(±1)^𝑛𝑞^{𝑛(𝑛𝑘−𝑙)/2}×𝑧^𝑛𝑓𝑞^−𝑛𝑇_𝑞⁻¹𝑇_𝑞^𝑛𝑘𝐸₂₂∶𝑓∈ℂ𝑤⁻¹,𝑤,𝒮𝔖^−±_{𝑞,𝑘,𝑙𝑛+1/2}=𝑧^𝑛𝑓𝑞^𝑛/2𝑇_𝑞𝐸₁₂+𝑧^𝑛+1𝑔𝑞^(𝑛+1)/2𝑇_𝑞×𝐸₂₁∶𝑓∈ℂ𝑤⁻¹,𝑤^{(𝛿𝑛,𝑛𝑘+𝑙)},∈ℂ𝑤⁻¹,𝑤^{(𝛿𝑛,(𝑛+1)𝑘−𝑙)}, (66) for all 𝑛∈ℤ.

Proof. It is easy to check that, Θ_−𝑠𝜎_{𝑎,𝑏,𝑐,𝑘,𝑙}Θ_𝑠=𝜎_{𝑎𝑞−𝑠𝑘,𝑏𝑞2𝑠,𝑐𝑞−𝑠𝑙,𝑘,𝑙}; (67) then; if we take 𝑠 such that 𝑞^2𝑠=𝑏⁻¹, we obtain the first assertion using (67) and the relations between 𝑎,𝑏,𝑐,𝑘,𝑙.
On the other hand, we suppose that 𝑧^𝑛𝑓(𝑇_𝑞)𝐸₁₁+𝑧^𝑛𝑔(𝑇_𝑞)𝐸₂₂∈(𝒮𝔖^+±_{𝑞,𝑘,𝑙})_𝑛; then, −𝑧^𝑛𝑓𝑇_𝑞𝐸₁₁−𝑧^𝑛𝑔𝑇_𝑞𝐸₂₂=𝜎^+±_{𝑞,𝑘,𝑙}𝑧^𝑛𝑓𝑇_𝑞𝐸₁₁+𝑧^𝑛𝑔𝑇_𝑞𝐸₂₂=(±1)^𝑛𝑞^{𝑛(𝑛𝑘−𝑙)/2}𝑧^𝑛𝑓𝑞^−𝑛𝑇_𝑞⁻¹𝑇_𝑞^𝑛𝑘𝐸₂₂+(±1)^𝑛𝑞^{𝑛(𝑛𝑘+𝑙)/2}𝑧^𝑛𝑔𝑞^−𝑛𝑇_𝑞⁻¹𝑇_𝑞^𝑛𝑘𝐸₁₁ (68) if and only if 𝑔(𝑇_𝑞)=−(±1)^𝑛𝑞^{𝑛(𝑛𝑘−𝑙)/2}𝑓(𝑞^−𝑛𝑇_𝑞⁻¹)𝑇_𝑞^𝑛𝑘. The proof for (𝒮𝔖^+±_{𝑞,𝑘,𝑙})_𝑛 is similar.
Now, by Theorem 1(b) and linearity of 𝜎⁺⁺_{𝑞,𝑘,𝑙}, we obtain that 𝑧^𝑛𝑓(𝑇_𝑞)𝐸₁₂+𝑧^𝑛+1𝑔(𝑇_𝑞)𝐸₂₁∈(𝒮𝔖⁺⁺_{𝑞,𝑘,𝑙})_𝑛+1/2 if and only if 𝑧^𝑛𝑓(𝑇_𝑞)𝐸₁₂ and 𝑧^𝑛𝑔(𝑇_𝑞)𝐸₂₁ are elements in (𝒮𝔖⁺⁺_{𝑞,𝑘,𝑙})_𝑛+1/2. We suppose that 𝑧^𝑛+1ℎ(𝑇_𝑞)𝐸₂₁∈(𝒮𝔖⁺⁺_{𝑞,𝑘,𝑙})_𝑛+1/2, with ℎ(𝑤)=∑ℎ_𝑗𝑤^𝑗; then, −𝑧^𝑛+1ℎ𝑇_𝑞𝐸₂₁=𝜎⁺⁺_𝑘,𝑙𝑧^𝑛+1ℎ𝑇_𝑞𝐸₂₁=−𝑞^{(𝑛+1)((𝑛+1)𝑘−𝑙)/2}𝑧^𝑛+1ℎ𝑞^−(𝑛+1)𝑇_𝑞⁻¹𝑇_𝑞^{(𝑛+1)𝑘−𝑙}𝐸₂₁ (69) if and only if  ℎ_𝑗𝑞^{−((𝑛+1)/2)𝑗}=𝑞^{−((𝑛+1)/2)((𝑛+1)𝑘−𝑙−𝑗)}ℎ_{(𝑛+1)𝑘−𝑙−𝑗}, for all 𝑗∈ℤ, which is equivalent to 𝑔(𝑤)=ℎ(𝑞^{−((𝑛+1)/2)}𝑤)∈ℂ[𝑤⁻¹,𝑤]^{(1,(𝑛+1)𝑘−𝑙)} (see Remark 8). Therefore, 𝑧^𝑛+1ℎ𝑇_𝑞𝐸₂₁=𝑧^𝑛+1𝑔𝑞^(𝑛+1)/2𝑇_𝑞𝐸₂₁,with𝑔(𝑤)∈ℂ𝑤⁻¹,𝑤^{(1,(𝑛+1)𝑘−𝑙)}. (70) Similarly, we prove that 𝑧^𝑛ℎ(𝑇_𝑞)𝐸₁₂∈(𝒮𝔖⁺⁺_{𝑞,𝑘,𝑙})_𝑛+1/2 if and only if 𝑧^𝑛ℎ(𝑇_𝑞)𝐸₁₂=𝑧^𝑛𝑓(𝑞^𝑛/2𝑇_𝑞)𝐸₁₂, with 𝑓∈ℂ[𝑤⁻¹,𝑤]^{(0,𝑛𝑘+𝑙)}.
The other proofs are similar.

Remark 11. From Lemma 9 (resp., Lemma 10), it is clear that an element in (𝒮𝔖_𝑞,𝑘)_𝑛+1/2 (resp., (𝒮𝔖^±,±_{𝑞,𝑘,𝑙})_𝑛) is totally determined by its value in the position 𝐸₁₂ (resp., 𝐸₁₁), where 𝑛∈ℤ.

Finally, we denote by 𝒮𝔖_𝑞,𝑘 and 𝒮𝔖^±±_{𝑞,𝑘,𝑙} the central extensions of 𝒮𝔖_𝑞,𝑘 and 𝒮𝔖^±±_{𝑞,𝑘,𝑙} corresponding to the restrictions of the 2-cocycle 𝜓, respectively. It is clear that these subalgebras admit a (1/2)ℤ-graduation compatible with their ℤ₂-graduation; that is, 𝒮𝔖_𝑞,𝑘=𝒮𝔖_𝑞,𝑘0𝒮𝔖_𝑞,𝑘1,𝒮𝔖^±±_{𝑞,𝑘,𝑙}=𝒮𝔖^±±_{𝑞,𝑘,𝑙0}𝒮𝔖^±±_{𝑞,𝑘,𝑙1}, (71) where 𝒮𝔖_𝑞,𝑘0=_𝑛∈ℤ𝒮𝔖_{𝑞,𝑘𝑛},𝒮𝔖_𝑞,𝑘1=_𝑛∈ℤ𝒮𝔖_{𝑞,𝑘𝑛+1/2},𝒮𝔖^±±_{𝑞,𝑘,𝑙0}=_𝑛∈ℤ𝒮𝔖^±±_{𝑞,𝑘,𝑙𝑛},𝒮𝔖^±±_{𝑞,𝑘,𝑙1}=_𝑛∈ℤ𝒮𝔖^±±_{𝑞,𝑘,𝑙𝑛+1/2}, (72) with (𝒮𝔖_𝑞,𝑘)_𝑛=(𝒮𝔖_𝑞,𝑘)_𝑛+𝛿_𝑛,0ℂ𝐶 and (𝒮𝔖^±±_{𝑞,𝑘,𝑙})_𝑛=(𝒮𝔖^±±_{𝑞,𝑘,𝑙})_𝑛+𝛿_𝑛,0ℂ𝐶, for all 𝑛∈(1/2)ℤ.

3. Quasifinite Highest Weight Modules over 𝒮𝔖_𝑞,𝑘 and 𝒮𝔖^±±_{𝑞,𝑘,𝑙}

The goal of this section is to characterize the quasifinite irreducible highest weight modules over 𝒮𝔖_𝑞,𝑘 and 𝒮𝔖^±±_{𝑞,𝑘,𝑙}; for this we will apply the general results on quasifinite representations of (1/2)ℤ-graded Lie superalgebras developed in Section 2 in [6]. Let us recall some general definition and results from [6].

In this section, 𝔤 denote a consistent (1/2)ℤ-graded Lie superalgebra over ℂ; namely, 𝔤=_{𝑗∈(1/2)ℤ}𝑔_𝑗,where𝔤_𝑖,𝔤_𝑗⊂𝔤_𝑖+𝑗1∀𝑖,𝑗∈2ℤ, (73) and also 𝔤₀=_𝑗∈ℤ𝔤_𝑗,𝔤₁=_𝑗∈ℤ𝔤_𝑗+1/2. (74) We denote 𝔤_±=⨁_{𝑗∈(1/2)ℤ+}𝔤_±𝑗. A subalgebra 𝔭 of 𝔤 is called parabolic if 𝔭=_{()ℤ𝑗∈1/2}𝔭_𝑗,where𝑝_𝑗=𝑔_𝑗𝔭∀𝑗⩾0,_−𝑗≠0,forsome𝑗∈ℕ. (75) We assume the following properties on 𝔤:(SP1)𝔤₀ is commutative,(SP2) if 𝑎∈𝔤_−𝑗 (𝑗∈(1/2)ℕ) and [𝑎,𝔤_1/2]=0, then 𝑎=0.Given that 𝑎∈𝔤_−1/2 is nonzero, we define 𝔭^𝑎=⨁_{𝑗∈(1/2)ℤ}𝔭^𝑎_𝑗, where 𝔭^𝑎_𝑗=𝔤_𝑗𝔭∀𝑗≥0,_𝑎−1/2=…𝑎,𝔤₀,𝔤₀,𝔭,…_{𝑎−𝑘−1/2}=𝔭^𝑎_−1/2,𝔭^𝑎_−𝑘. (76) It was proved in [6] that 𝔭^𝑎 is the minimal parabolic subalgebra containing 𝑎 and also that [𝔭^𝑎,𝔭^𝑎]∩𝔤₀=[𝑎,𝔤_1/2].

Definition 12. (a) A parabolic subalgebra 𝔭 is called nondegenerate if 𝔭_−𝑗 has finite codimension in 𝔤_−𝑗, for all 𝑗∈(1/2)ℕ.
(b) An element 𝑎∈𝔤_−1/2 is called nondegenerate if 𝔭^𝑎 is nondegenerate.

We will also require the following condition on 𝔤.(SP3) If 𝔭 is a nondegenerate parabolic subalgebra of 𝔤, then there exists a nondegenerate element 𝑎∈𝔭_−1/2.

A 𝔤-module 𝑉=⨁_{𝑗∈(1/2)ℤ}𝑉_𝑗 is called quasifinite if dim𝑉_𝑗<∞ for all 𝑗. Given 𝜆∈𝔤^∗₀, a highest weight module is a (1/2)ℤ-graded 𝔤-module 𝑉(𝔤,𝜆)=⨁_{𝑗∈(1/2)ℤ+}𝑉_−𝑗, defined by the following properties: (a)𝑉₀=ℂ𝑣_𝜆, where 𝑣_𝜆 is a nonzero vector; (b)ℎ𝑣_𝜆=𝜆(ℎ)𝑣_𝜆, for all ℎ∈𝔤₀; (c)𝔤₊𝑣_𝜆=0; (d)𝒰(𝔤₋)𝑣_𝜆=𝑉(𝔤,𝜆).

A nonzero vector 𝑣∈𝑉(𝔤,𝜆) is called singular if 𝔤₊𝑣=0.

The Verma module over 𝔤 is defined as usual: 𝑀(𝔤,𝜆)=𝒰(𝔤)_{𝒰𝔤0⨁𝔤+}ℂ_𝜆, (77) where ℂ_𝜆∶=ℂ𝑐_𝜆 is the one-dimensional (𝔤₀⨁𝔤₊)-module given by ℎ𝑐_𝜆=𝜆(ℎ)𝑐_𝜆 if ℎ∈𝔤₀, and 𝔤₊𝑐_𝜆=0 and the action of 𝔤 on 𝑀(𝔤,𝜆) is induced by the left multiplication in 𝒰(𝔤). Any highest weight module 𝑉(𝔤,𝜆) is a quotient module of 𝑀(𝔤,𝜆). The irreducible module 𝐿(𝔤,𝜆) is the quotient of 𝑀(𝔤,𝜆) by the maximal proper graded submodule.

Now, let 𝔭=⨁_{𝑗∈(1/2)ℤ}𝔭_𝑗 be a parabolic subalgebra of 𝔤 and let 𝜆∈𝔤^∗₀ be such that 𝜆|_{[𝔭,𝔭]∩𝔤0}=0. Then the (𝔤₀⨁𝔤₊)-module ℂ_𝜆 extends to a 𝔭-module by letting 𝔭_𝑗𝑐_𝜆=0 for all 𝑗<0, and we may construct the highest weight module 𝑀(𝔭,𝔤,𝜆)=𝒰(𝔤)_𝒰(𝔭)ℂ_𝜆, (78) 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 𝔤=⨁_{𝑗∈(1/2)ℤ}𝔤_𝑗 be a consistent (1/2)ℤ-graded Lie superalgebra over ℂ that satisfies conditions (SP1), (SP2), and (SP3). The following conditions on 𝜆∈𝔤^∗₀ are equivalent.(a)𝑀(𝔤,𝜆) contains a singular vector 𝑎⋅𝑣_𝜆 in 𝑀(𝔤,𝜆)_−1/2, where 𝑎 is nondegenerate. (b)There exists a nondegenerate element 𝑎∈𝔤_−1/2, such that 𝜆([𝔤_1/2,𝑎])=0. (c)𝐿(𝔤,𝜆) is quasifinite. (d)There exists a nondegenerate element 𝑎∈𝔤_−1/2, such that 𝐿(𝔤,𝜆) is the irreducible quotient of the generalized Verma module 𝑀(𝔤,𝔭^𝑎,𝜆).

Proof. See [6].

Moreover, we will need the following result. Recall that a quasipolynomial is a combination of functions of the form 𝑝(𝑥)𝑞^𝛼𝑥, where 𝑝(𝑥) is a polynomial and 𝛼∈ℂ. That is, it satisfies a nontrivial linear differential equation with constant coefficients.

Proposition 14. Given a quasipolynomial 𝑃 and a polynomial 𝐵(𝑥)=∏_𝑖(𝑥−𝐴_𝑖), take 𝑏(𝑥)=∏_𝑖(𝑥−𝑎_𝑖) where 𝑎_𝑖=exp(𝐴_𝑖); then, 𝑏(𝑥)(∑_𝑛∈ℤ𝑃(𝑛)𝑥^−𝑛)=0 if and only if 𝐵(𝑑/𝑑𝑥)𝑃(𝑥)=0.

Below, we prove that 𝒮𝔖_𝑞,𝑘 and 𝒮𝔖^±±_{𝑞,𝑘,𝑙} 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 𝐴=𝑧^𝑛𝑓₁(𝑇_𝑞)𝐸₁₁+𝑧^𝑛𝑓₂(𝑇_𝑞)𝐸₂₂ and 𝑧^𝑚𝑔(𝑇_𝑞)𝐸_𝑖𝑗 be nonzero elements in 𝒮𝔖_𝑞, where 𝑛≠0,  𝑖,𝑗=1,2 and let ℎ∈ℂ[𝑤⁻¹,𝑤] be nonconstant. Then one has the following.(a)[𝐴,𝑧^𝑚𝑔(𝑇_𝑞)𝐸_𝑖𝑗]≠0 or [𝐴,𝑧^𝑚𝑔(𝑇_𝑞)ℎ(𝑇_𝑞)𝐸_𝑖𝑗]≠0, with 𝑖≠𝑗. (b)If 𝑓_𝑖≠0, then [𝐴,𝑧^𝑚𝑔(𝑇_𝑞)𝐸_𝑖𝑖]≠0 or [𝐴,𝑧^𝑚𝑔(𝑇_𝑞)ℎ(𝑇_𝑞)𝐸_𝑖𝑖]≠0.

Proof. (a) We suppose that [𝐴,𝑧^𝑚𝑔(𝑇_𝑞)𝐸₁₂]=0 and [𝐴,𝑧^𝑚𝑔(𝑇_𝑞)ℎ(𝑇_𝑞)𝐸₁₂]=0; then, 𝑓₁𝑞^𝑚𝑇_𝑞𝑔𝑇_𝑞=𝑓₂𝑇_𝑞𝑔𝑞^𝑛𝑇_𝑞𝑓₁,𝑞^𝑚𝑇_𝑞𝑔𝑇_𝑞ℎ𝑇_𝑞=𝑓₂𝑇_𝑞𝑔𝑞^𝑛𝑇_𝑞ℎ𝑞^𝑛𝑇_𝑞. (79)(80) Using (79) and the hypothesis, we have that 𝑓₁ and 𝑓₂ are nonzero. Then, if we replace (79) with (80), we obtain ℎ(𝑇_𝑞)=ℎ(𝑞^𝑛𝑇_𝑞) which is a contradiction since ℎ is not constant, 𝑛≠0, and 𝑞 is not unity root. The other case is similar.
The proof of (b) is similar to the proof of (a).

3.1. The Case 𝔤=𝒮𝔖_𝑞,𝑘

It is clear that 𝒮𝔖_𝑞,𝑘 satisfies (SP1). Now, we suppose that 𝐴=𝑧^−𝑛𝑓(𝑇_𝑞)𝐸₁₁+𝑧^−𝑛𝑔(𝑇_𝑞)𝐸₂₂∈(𝒮𝔖_𝑞,𝑘)_−𝑛 (with 𝑛∈ℕ) satisfies [𝐴,(𝒮𝔖_𝑞,𝑘)_1/2]=0; in particular for all 𝑖∈ℤ𝐴,𝑇^𝑖_𝑞𝐸₁₂−𝑧𝑇_𝑞^𝑘−𝑖𝐸₂₁=0. (81) Then using (81), we obtain that 𝑓(𝑇_𝑞)=𝑞^−𝑛𝑖𝑔(𝑇_𝑞) for all 𝑖∈ℤ. Hence, 𝑓=𝑔=0 (since 𝑛≠0 and 𝑞 is not a root of unity); therefore, 𝐴=0. Now, we suppose that 𝐴=𝑧^−𝑛𝑓(𝑇_𝑞)𝐸₁₂−(−1)^−𝑛𝑞^{(𝑛−1)𝑛𝑘}𝑧^−𝑛+1𝑓(𝑞^𝑛𝑇_𝑞⁻¹)𝑇_𝑞^{(−2𝑛+1)𝑘}𝐸₂₁ in (𝒮𝔖_𝑞,𝑘)_−𝑛+1/2; (with 𝑛∈ℕ) satisfies [𝐴,(𝒮𝔖_𝑞,𝑘)_1/2]=0 then, for all 𝑖∈ℤ we have that 𝐴,𝑇^𝑖_𝑞𝐸₁₂−𝑧𝑇_𝑞^𝑘−𝑖𝐸₂₁=0, (82) which is equivalent to 𝑓(𝑇_𝑞)=𝑞^{𝑛𝑘−𝑖}𝑓(𝑞𝑇_𝑞) for all 𝑖∈ℤ. Hence 𝑓=0; therefore, 𝐴=0. Thus, we prove that 𝒮𝔖_𝑞,𝑘 satisfies the property (SP2).

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

Lemma 16. Let 𝔭=⨁_{𝑗∈(1/2)ℤ}𝔭_𝑗 be a (1/2)ℤ-graded subalgebra of 𝒮𝔖_𝑞,𝑘, where 𝔭₀=(𝒮𝔖_𝑞,𝑘)₀. Then one has the following. (a)For each 𝑛∈ℤ, 𝔭_𝑛+1/2 has finite codimension in (𝒮𝔖_𝑞,𝑘)_𝑛+1/2 if and only if 𝔭_𝑛+1/2≠0. (b)For each 𝑛∈ℤ, 𝔭_𝑛 has finite codimension in (𝒮𝔖_𝑞,𝑘)_𝑛 if and only if there exists 𝑧^𝑛𝑓(𝑞^𝑛/2𝑇_𝑞)𝐸₁₁+𝑧^𝑛𝑔(𝑞^(𝑛−1)/2𝑇_𝑞)𝐸₂₂∈𝔭_𝑛, such that 𝑓 and 𝑔 are nonzero. (c)𝔭_−𝑛 has finite codimension in (𝒮𝔖_𝑞,𝑘)_−𝑛, for all 𝑛∈(1/2)ℕ if and only if 𝔭_−1/2≠0.

Proof. (a) We suppose that there exists 𝐴=𝑧^𝑛𝑓𝑇_𝑞𝐸₁₂−(−1)^𝑛𝑞^{(𝑛+1)𝑛𝑘}𝑧^𝑛+1×𝑓𝑞^−𝑛𝑇_𝑞⁻¹𝑇_𝑞^{(2𝑛+1)𝑘}𝐸₂₁∈𝔭_𝑛+1/2, (83) with 𝑓≠0; then 𝑀^𝑗₁∶=[𝐴,(𝑇^𝑗_𝑞−𝑇_𝑞^−𝑗)𝐸₁₁] and 𝑀^𝑗₂∶=[𝐴,(𝑞^−𝑗/2𝑇^𝑗_𝑞−𝑞^𝑗/2𝑇_𝑞^−𝑗)𝐸₂₂] belong to 𝔭_𝑛+1/2, for all 𝑗∈ℤ. Moreover, 𝐴^𝑗=(𝑞^{−(𝑛+1)𝑗}−𝑞^𝑛𝑗)⁻¹(𝑀^𝑗₁+𝑞^{−(𝑛+1/2)𝑗}𝑀^𝑗₂)∈𝔭_𝑛+1/2 and 𝐴^𝑗=𝑧^𝑛𝑓𝑇_𝑞𝑇^𝑗_𝑞𝐸₁₂−(−1)^𝑛𝑞^{(𝑛+1)𝑛𝑘−𝑛𝑗}𝑧^𝑛+1×𝑓𝑞^−𝑛𝑇_𝑞⁻¹𝑇_𝑞^{(2𝑛+1)𝑘−𝑗}𝐸₂₁, (84) for all 𝑗∈ℤ^×. Therefore, using (84), we have that 𝑧^𝑛𝑔𝑇_𝑞𝐸₁₂−(−1)^𝑛𝑞^{(𝑛+1)𝑛𝑘}𝑧^𝑛+1×𝑔𝑞^−𝑛𝑇_𝑞⁻¹𝑇_𝑞^{(2𝑛+1)𝑘}𝐸₂₁∈𝔭_𝑛+1/2, (85) for all 𝑔(𝑤)∈⟨𝑓(𝑤)⟩, where ⟨𝑓(𝑤)⟩ is the ideal of ℂ[𝑤⁻¹,𝑤] generated by 𝑓(𝑤). Using that ⟨𝑓(𝑤)⟩ has finite codimension in ℂ[𝑤⁻¹,𝑤], we obtain that 𝔭_𝑛+1/2 has finite codimension in (𝒮𝔖_𝑞,𝑘)_𝑛+1/2 (see Remark 11). The proof of the converse is trivial.
(b) We suppose that 𝐴=𝑧^𝑛𝑓(𝑞^𝑛/2𝑇_𝑞)𝐸₁₁+𝑧^𝑛𝑔(𝑞^(𝑛−1)/2𝑇_𝑞)𝐸₂₂∈𝔭_𝑛, with nonzero 𝑓 and 𝑔 in ℂ[𝑤⁻¹,𝑤]^{(𝑛,2𝑛𝑘)}; then similar to the aforementioned argument 𝑞^{−𝑛𝑗/2}−𝑞^{𝑛𝑗/2−1}𝐴,𝑇^𝑗_𝑞−𝑇_𝑞^−𝑗𝐸₁₁=𝑧^𝑛𝑓𝑞^𝑛/2𝑇_𝑞𝑞^𝑛𝑗/2𝑇^𝑗_𝑞+𝑞^{−𝑛𝑗/2}𝑇_𝑞^−𝑗𝐸₁₁∈𝔭_𝑛,𝑏𝐴,𝑞^−𝑗/2𝑇^𝑗_𝑞−𝑞^𝑗/2𝑇_𝑞^−𝑗𝐸₂₂=𝑧^𝑛𝑔𝑞^(𝑛−1)/2𝑇_𝑞×𝑞^{(𝑛−1)𝑗/2}𝑇^𝑗_𝑞+𝑞^{−(𝑛−1)𝑗/2}𝑇_𝑞^−𝑗𝐸₂₂∈𝔭_𝑛, (86) where 𝑏=(𝑞^{(−𝑛/2)𝑗}−𝑞^(𝑛/2)𝑗)⁻¹, and this is true for all 𝑗∈ℤ^×. Therefore 𝑧^𝑛ℎ₁𝑞^𝑛/2𝑇_𝑞𝐸₁₁∈𝔭_𝑛,∀ℎ₁∈𝑓(𝑤)ℂ𝑤⁻¹,𝑤^(1,0),𝑧^𝑛ℎ₂𝑞^(𝑛−1)/2𝑇_𝑞𝐸₂₂∈𝔭_𝑛,∀ℎ₂∈𝑔(𝑤)ℂ𝑤⁻¹,𝑤^(1,0). (87) Since 𝑓(𝑤)ℂ[𝑤⁻¹,𝑤]^(1,0) and 𝑔(𝑤)ℂ[𝑤⁻¹,𝑤]^(1,0) have finite codimension in ℂ[𝑤⁻¹,𝑤]^{(𝑛,2𝑛𝑘)}, we obtain that 𝔭_𝑛 has finite codimension in (𝒮𝔖_𝑞,𝑘)_𝑛. The proof of the converse is trivial.
(c) We suppose that 𝔭_−1/2≠0; then, in order to prove that 𝔭_−𝑛 has finite codimension in (𝒮𝔖_𝑞,𝑘)_−𝑛, for all 𝑛∈(1/2)ℕ, we only need to see that this is true for all 𝑛∈ℕ, since by using (SP2), we obtain that 𝔭_−𝑛+1/2≠0 for all 𝑛∈ℕ; then, from (a) we have that 𝔭_−𝑛+1/2 has finite codimension in (𝒮𝔖_𝑞,𝑘)_−𝑛+1/2 for all 𝑛∈ℕ. By hypothesis, there exists 𝐴=𝑧⁻¹𝑓(𝑇_𝑞)𝐸₁₂+𝑓(𝑞𝑇_𝑞⁻¹)𝑇_𝑞^−𝑘𝐸₂₁∈𝔭_−1/2 with 𝑓≠0; then, using (84), we have that 𝐵=𝑧⁻¹𝑓(𝑇_𝑞)𝑇_𝑞𝐸₁₂+𝑞𝑓(𝑞𝑇_𝑞⁻¹)𝑇_𝑞^−𝑘−1𝐸₂₁∈𝔭_−1/2, and [𝐴,𝐵]∶=𝑧⁻¹𝑓₁𝑞^−1/2𝑇_𝑞𝐸₁₁+𝑧⁻¹𝑔₁𝑞⁻¹𝑇_𝑞𝐸₂₂∈𝔭₋₁, (88) where 𝑓₁ and 𝑔₁ are nonzero. Hence, by (b), 𝔭₋₁ has finite codimension in (𝒮𝔖_𝑞,𝑘)₋₁. Moreover, by (87) we obtain that 𝑧⁻¹𝑓₁𝑞^−1/2𝑇_𝑞ℎ𝑞^−1/2𝑇_𝑞𝐸₁₁∈𝔭₋₁,𝑧⁻¹𝑔₁𝑞⁻¹𝑇_𝑞ℎ𝑞⁻¹𝑇_𝑞𝐸₂₂∈𝔭₋₁, (89)(90) for all ℎ(𝑤)∈ℂ[𝑤⁻¹,𝑤]^(1,0). Now, by induction we suppose that 𝔭_−𝑛 has finite codimension in (𝒮𝔖_𝑞,𝑘)_−𝑛; then, there exists 𝐴=𝑧^−𝑛𝑓(𝑞^−𝑛/2𝑇_𝑞)𝐸₁₁+𝑧^−𝑛𝑔(𝑞^{−(𝑛+1)/2}𝑇_𝑞)𝐸₂₂∈𝔭_−𝑛 where 𝑓 and 𝑔 are nonzero, and by (89) and Lemma 15(b), 𝐴,𝑧⁻¹𝑓₁𝑞^−1/2𝑇_𝑞𝐸₁₁≠0or𝐴,𝑧⁻¹𝑓₁𝑞^−1/2𝑇_𝑞ℎ𝑞^−1/2𝑇_𝑞𝐸₁₁≠0, (91) for some no constant ℎ(𝑤) in ℂ[𝑤⁻¹,𝑤]^(1,0); therefore, there exists a nonzero element 𝑧^−𝑛−1𝑓(𝑇_𝑞)𝐸₁₁ in 𝔭_−𝑛−1. Similarly, using (90) and Lemma 15(b), we see that there exists a nonzero element 𝑧^−𝑛−1𝑔(𝑇_𝑞)𝐸₂₂ in 𝔭_−𝑛−1; then, using (b), 𝔭_−𝑛−1 has finite codimension in (𝒮𝔖_𝑞,𝑘)_−𝑛−1, finishing the induction. The proof of the converse is trivial.

Corollary 17. (a) Any parabolic subalgebra of 𝒮𝔖_𝑞,𝑘 is nondegenerate.
(b) Any nonzero element of (𝒮𝔖_𝑞,𝑘)_−1/2 is nondegenerate.
(c) 𝒮𝔖_𝑞,𝑘 satisfies (SP3).

Proof. Let 𝔭 be a parabolic subalgebra of 𝒮𝔖_𝑞,𝑘; by definition there exists 𝑗∈(1/2)ℕ such that 𝔭_−𝑗≠0, and then by (SP2) 𝔭_−1/2≠0; the proof of (a) follows from Lemma 16(c). Finally, (b) follows from (a), and (c) follows from (b).

A functional 𝜆∈(𝒮𝔖_𝑞,𝑘)^∗₀ is described by its labels Δ_𝑛,1=−𝜆((𝑇^𝑛_𝑞−𝑇_𝑞^−𝑛)𝐸₁₁), Δ_𝑛,2=−𝜆((𝑞^−𝑛𝑇^𝑛_𝑞−𝑇_𝑞^−𝑛)𝐸₂₂) with 𝑛∈ℤ, and the central charge 𝜆(𝐶)=𝑐. We will consider the generating series Δ_𝜆,𝑖(𝑥)=_𝑛∈ℤΔ_𝑛,𝑖𝑥^−𝑛,with𝑖=1,2. (92)

Theorem 18. An irreducible highest weight 𝒮𝔖_𝑞,𝑘-module 𝐿(𝒮𝔖_𝑞,𝑘,𝜆) is quasifinite if and only if one of the following equivalent conditions holds. (a)There exists a Laurent polynomial 𝑏(𝑥) such that 𝑏𝑞𝑥Δ_𝜆,1(𝑥)+Δ_𝜆,2(𝑥)+𝑐=0. (93)(b)There exists a quasipolynomial 𝑃(𝑥) such that 𝑃(𝑛)=Δ_𝑛,1+Δ_𝑛,2+𝛿_𝑛,0𝑐, (94) for all 𝑛∈ℤ.

Proof. By Theorem 13, 𝐿(𝒮𝔖_𝑞,𝑘,𝜆) is quasifinite if and only if there exist a nondegenerate element 𝑎 in (𝒮𝔖_𝑞,𝑘)_−1/2, such that 𝜆([(𝒮𝔖_𝑞,𝑘)_1/2,𝑎])=0.
Now, let 𝑎=𝑧⁻¹𝑏(𝑇_𝑞)𝐸₁₂+𝑏(𝑞𝑇_𝑞⁻¹)𝑇_𝑞^−𝑘𝐸₂₁ be a nonzero element of (𝒮𝔖_𝑞,𝑘)_−1/2 (with 𝑏(𝑤)=∑_𝑗𝑏_𝑗𝑤^𝑗); then, by Corollary 17(b), 𝑎 is a nondegenerate element. Then, 𝜆([(𝒮𝔖_𝑞,𝑘)_1/2,𝑎])=0 if and only if, for all 𝑖∈ℤ, 0=𝜆𝑇^𝑖_𝑞𝐸₁₂−𝑧𝑇_𝑞^𝑘−𝑖𝐸₂₁=,𝑎_𝑗𝑞^𝑗𝑏_𝑗Δ_{𝑘−𝑖+𝑗,1}+Δ_{𝑘−𝑖+𝑗,2}+𝑞^−𝑘+𝑖𝑏_−𝑘+𝑖𝑐. (95) Multiplying (95) by 𝑥^−𝑘+𝑖 and adding over 𝑖∈ℤ, we obtain that 0=_𝑗,𝑖𝑞^𝑗𝑏_𝑗Δ_{𝑘−𝑖+𝑗,1}+Δ_{𝑘−𝑖+𝑗,2}𝑥^−𝑘+𝑖+_𝑖𝑞^−𝑘+𝑖𝑏_−𝑘+𝑖𝑥^−𝑘+𝑖𝑐=𝑏𝑞𝑥Δ_𝜆,1(𝑥)+Δ_𝜆,2(𝑥)+𝑐. (96) The equivalence between (a) and (b) follows from Proposition 14.

3.2. The Case 𝔤=𝒮𝔖^±±_{𝑞,𝑘,𝑙}

It is clear that 𝒮𝔖^±±_{𝑞,𝑘,𝑙} satisfies (SP1). Now, we prove that 𝒮𝔖^±±_{𝑞,𝑘,𝑙} satisfies (SP2). We suppose that 𝐴=𝑧^−𝑛𝑓(𝑞^−𝑛/2𝑇_𝑞)𝐸₁₂+𝑧^−𝑛+1𝑔(𝑞^{(−𝑛+1)/2}𝑇_𝑞)𝐸₂₁ in (𝒮𝔖^±±_{𝑞,𝑘,𝑙})_−𝑛+1/2 (with 𝑛∈ℕ) satisfies [𝐴,(𝒮𝔖^±±_{𝑞,𝑘,𝑙})_1/2]=0; in particular if we take nonzero elements ℎ₁₂(𝑇_𝑞)𝐸₁₂ and 𝑧ℎ₂₁(𝑞^1/2𝑇_𝑞)𝐸₂₁ in (𝒮𝔖^±±_{𝑞,𝑘,𝑙})_1/2, we have that 𝐴,ℎ₁₂𝑇_𝑞𝐸₁₂=0; (97) that is, 𝑔(𝑞^{(−𝑛+1)/2}𝑇_𝑞)ℎ₁₂(𝑇_𝑞)=0; therefore, 𝑔(𝑤)=0. Similarly, 𝐴,𝑧ℎ₂₁𝑞^1/2𝑇_𝑞𝐸₂₁=0, (98) which is equivalent to 𝑓(𝑞^−𝑛/2+1𝑇_𝑞)ℎ₂₁(𝑞^1/2𝑇_𝑞)=0; therefore, 𝑓(𝑤)=0, and then 𝐴=0.

Remark 19. Given 𝐵=𝑧^𝑛𝑓(𝑞^𝑛/2𝑇_𝑞)𝐸₁₂+𝑧^𝑛+1𝑔(𝑞^(𝑛+1)/2𝑇_𝑞)𝐸₂₁∈(𝒮𝔖^±±_{𝑞,𝑘,𝑙})_𝑛+1/2 and 𝐴=ℎ(𝑇_𝑞)𝐸₁₁−ℎ(𝑇_𝑞⁻¹)𝐸₂₂∈(𝒮𝔖^±±_{𝑞,𝑘,𝑙})₀ with ℎ∈ℂ[𝑤⁻¹,𝑤], then 𝑀∶=[𝐴,𝐵]∈(𝒮𝔖^±±_{𝑞,𝑘,𝑙})_𝑛+1/2, where 𝑀_𝑖𝑖=0(𝑖=1,2) and 𝑀₁₂=−𝑧^𝑛𝑓𝑞^𝑛/2𝑇_𝑞ℎ𝑞^𝑛𝑇_𝑞+ℎ𝑇_𝑞⁻¹,𝑀₂₁=𝑧^𝑛+1𝑔𝑞^(𝑛+1)/2𝑇_𝑞ℎ𝑇_𝑞+ℎ𝑞^−𝑛−1𝑇_𝑞⁻¹. (99)(100)

Now, let 𝐴=𝑧^−𝑛𝑓₁(𝑇_𝑞)𝐸₁₁+𝑧^−𝑛𝑓₂(𝑇_𝑞)𝐸₂₂∈(𝒮𝔖^±±_{𝑞,𝑘,𝑙})_−𝑛 (with 𝑛∈ℕ) be such that 𝐴 is nonzero; then, if we take nonzero 𝐵=𝑓(𝑇_𝑞)𝐸₁₂ in (𝒮𝔖^±±_{𝑞,𝑘,𝑙})_1/2 by Remark 19, there exists 𝑓(𝑇_𝑞)ℎ(𝑇_𝑞)𝐸₁₂∈(𝒮𝔖^±±_{𝑞,𝑘,𝑙})_1/2 such that ℎ is not constant; then by Lemma 15(a), we have that 𝐴,𝑓𝑇_𝑞𝐸₁₂≠0or𝐴,𝑓𝑇_𝑞ℎ𝑇_𝑞𝐸₁₂≠0. (101) Therefore 𝒮𝔖^±±_{𝑞,𝑘,𝑙} satisfies (SP2).

In order to prove that 𝒮𝔖^±±_{𝑞,𝑘,𝑙} satisfies (SP3), we will need the following result. We denote 𝛿^′_𝑛𝛿∶=^𝑛+1,in𝒮𝔖^+,±_{𝑞,𝑘,𝑙𝑛+1/2},𝛿_𝑛,in𝒮𝔖^−,±_{𝑞,𝑘,𝑙𝑛+1/2}. (102)

Lemma 20. Let 𝔭=⨁_{𝑗∈(1/2)ℤ}𝔭_𝑗 be a (1/2)ℤ-graded subalgebra of 𝒮𝔖^±±_{𝑞,𝑘,𝑙}, with 𝔭₀=(𝒮𝔖^±±_{𝑞,𝑘,𝑙})₀. Then one has the following: (a)for each 𝑛∈ℤ,  𝔭_𝑛 has finite codimension in (𝒮𝔖^±±_{𝑞,𝑘,𝑙})_𝑛 if and only if 𝔭_𝑛≠0; (b)for each 𝑛∈ℤ,  𝔭_𝑛+1/2 has finite codimension in (𝒮𝔖^±±_{𝑞,𝑘,𝑙})_𝑛+1/2 if and only if there exists 𝑧^𝑛𝑓(𝑞^𝑛𝑇_𝑞)𝐸₁₂+𝑧^𝑛+1𝑔(𝑞^(𝑛+1)/2𝑇_𝑞)𝐸₂₁∈𝔭_𝑛+1/2, such that 𝑓 and 𝑔 are nonzero; (c)𝔭_−𝑛 has finite codimension in (𝒮𝔖^±±_{𝑞,𝑘,𝑙})_−𝑛 for all 𝑛∈(1/2)ℕ, if and only if 𝔭_−1/2 has finite codimension in (𝒮𝔖^±±_{𝑞,𝑘,𝑙})_−1/2.

Proof. (a) Let 𝐴=𝑧^𝑛𝑓(𝑇_𝑞)𝐸₁₁−(±1)^𝑛𝑞^{𝑛(𝑛𝑘−𝑙)/2}𝑧^𝑛𝑓(𝑞^−𝑛𝑇_𝑞⁻¹)𝑇_𝑞^𝑛𝑘𝐸₂₂∈𝔭_𝑛, with 𝑓≠0; then, 𝐵_𝑗∶=(1−𝑞^𝑛𝑗)⁻¹[𝐴,𝑇^𝑗_𝑞𝐸₁₁−𝑇_𝑞^−𝑗𝐸₂₂]∈𝔭_𝑛 and 𝐵_𝑗=𝑧^𝑛𝑓𝑇_𝑞𝑇^𝑗_𝑞𝐸₁₁−(±1)^𝑛𝑞^{(𝑛(𝑛𝑘−𝑙)/2)−𝑛𝑗}×𝑧^𝑛𝑓𝑞^−𝑛𝑇_𝑞⁻¹𝑇_𝑞^{𝑛𝑘−𝑗}𝐸₂₂, (103) for all 𝑗∈ℤ^×; therefore, by (103) we have that 𝑧^𝑛𝑔𝑇_𝑞𝐸₁₁−(±1)^𝑛𝑞^{𝑛(𝑛k−𝑙)/2}𝑧^𝑛𝑔𝑞^−𝑛𝑇_𝑞⁻¹𝑇_𝑞^𝑛𝑘𝐸₂₂∈𝔭_𝑛, (104) for all 𝑔(𝑤)∈⟨𝑓(𝑤)⟩, and since ⟨𝑓(𝑤)⟩ has finite codimension in ℂ[𝑤⁻¹,𝑤], we have that 𝔭_𝑛 has finite codimension in (𝒮𝔖^±±_{𝑞,𝑘,𝑙})_𝑛 (see Remark 11). The proof of the converse is trivial.
(b) We suppose that there exist nonzero elements 𝑧^𝑛𝑓₁𝑇_𝑞𝐸₁₂,𝑧^𝑛+1𝑓₂𝑇_𝑞𝐸₂₁in𝔭_𝑛+1/2; (105) then, using Remark 19, we obtain that 𝑧^𝑛𝑔₁(𝑇_𝑞)𝐸₁₂∈𝔭_𝑛+1/2 for all 𝑔₁(𝑞^−𝑛/2𝑤)∈𝑓₁(𝑤)ℂ[𝑤⁻¹,𝑤]^(1,0) (see (99)) and 𝑧^𝑛+1𝑔₂(𝑇_𝑞)𝐸₂₁∈𝔭_𝑛+1/2 for all 𝑔₂(𝑞^{(−𝑛+1)/2}𝑤)∈𝑓₂(𝑤)ℂ[𝑤⁻¹,𝑤]^(1,0) (see (100)). Then using that 𝑓₁(𝑤)ℂ[𝑤⁻¹,𝑤]^(1,0) has finite codimension in ℂ[𝑤⁻¹,𝑤]^{(𝛿𝑛,𝑛𝑘+𝑙)} and 𝑓₂(𝑤)ℂ[𝑤⁻¹,𝑤]^(1,0) has finite codimension in ℂ[𝑤⁻¹,𝑤]^{(𝛿′𝑛,(𝑛+1)𝑘−𝑙)}, we obtain that 𝔭_𝑛+1/2 has finite codimension in (𝒮𝔖^±±_{𝑞,𝑘,𝑙})_𝑛+1/2. Therefore in order to prove that 𝔭_𝑛+1/2 has finite codimension in (𝒮𝔖^±±_{𝑞,𝑘,𝑙})_𝑛+1/2, we only need to see that there exist nonzero elements as aforementioned in 𝔭_𝑛+1/2. Let 𝐵=𝑧^𝑛𝑓(𝑞^𝑛/2𝑇_𝑞)𝐸₁₂+𝑧^𝑛+1𝑔(𝑞^(𝑛+1)/2𝑇_𝑞)𝐸₂₁∈𝔭_𝑛+1/2 with nonzero 𝑓 and 𝑔, and let 𝐴 and 𝑀 be as in Remark 19. Then, taking ℎ(𝑤)=𝑤−𝑞^𝑛𝑤⁻¹ (observe that ℎ(𝑇_𝑞⁻¹)=−ℎ(𝑞^𝑛𝑇_𝑞)), we obtain that 𝑀=𝑧^𝑛+1𝑔𝑞^(𝑛+1)/2𝑇_𝑞×ℎ𝑇_𝑞+ℎ𝑞^−(𝑛+1)𝑇_𝑞⁻¹×𝐸₂₁∈𝔭_𝑛+1/2,with𝑀≠0. (106) Similarly, taking ℎ(𝑤)=𝑤−𝑞^−𝑛−1𝑤⁻¹ (observe that ℎ(𝑇_𝑞)=−ℎ(𝑞^−𝑛−1𝑇_𝑞⁻¹)), we obtain 𝑀=−𝑧^𝑛𝑓𝑞^𝑛/2𝑇_𝑞×ℎ𝑞^𝑛𝑇_𝑞+ℎ𝑇_𝑞⁻¹𝐸₁₂∈𝔭_𝑛+1/2,with𝑀≠0, (107) proving the existence of nonzero elements of those forms. The proof of the converse is trivial.
(c) We suppose that 𝔭_−1/2 has finite codimension in (𝒮𝔖^±±_{𝑞,𝑘,𝑙})_−1/2; then, in order to prove that 𝔭_−𝑛 has finite codimension in (𝒮𝔖^±±_{𝑞,𝑘,𝑙})_−𝑛 for all 𝑛∈(1/2)ℕ, we only need to see that this is true for all −𝑛+1/2 with 𝑛∈ℕ, since in this case by using (SP2), we obtain that 𝔭_−𝑛≠0 for all 𝑛∈ℕ; then, by (a), we have that 𝔭_−𝑛 has finite codimension in (𝒮𝔖^±±_{𝑞,𝑘,𝑙})_−𝑛, for all 𝑛∈ℕ.
By hypothesis, there exist nonzero elements 𝐴=𝑧⁻¹𝑓(𝑞⁻¹𝑇_𝑞)𝐸₁₂ and 𝐵=𝑔(𝑇_𝑞)𝐸₂₁ in 𝔭_−1/2; therefore, [𝐴,𝐵]∈𝔭₋₁ is nonzero. By induction, we suppose that 𝔭_−𝑛+1/2 has finite codimension in (𝒮𝔖^±±_{𝑞,𝑘,𝑙})_−𝑛+1/2; then, there exist nonzero elements 𝑧^−𝑛𝑓₁₂(𝑞^𝑛𝑇_𝑞)𝐸₁₂ and 𝑧^−𝑛+1𝑓₂₁(𝑞^{(−𝑛+1)/2}𝑇_𝑞)𝐸₂₁ in (𝒮𝔖^±±_{𝑞,𝑘,𝑙})_−𝑛+1/2, and by Remark 19, we also have 𝑧^−𝑛𝑓₁₂(𝑞^𝑛𝑇_𝑞)𝑔₁₂(𝑇_𝑞)𝐸₁₂, 𝑧^−𝑛+1𝑓₂₁(𝑞^{(−𝑛+1)/2}𝑇_𝑞)𝑔₂₁(𝑇_𝑞)𝐸₂₁ in (𝒮𝔖^±±_{𝑞,𝑘,𝑙})_−𝑛+1/2 with no constant 𝑔_𝑖𝑗 for all 𝑖≠𝑗. Then by Lemma 15(a), we have that 𝑧^−𝑛𝑓₁₂𝑞^𝑛𝑇_𝑞𝐸₁₂,[𝑧𝐴,𝐵]≠0or^−𝑛𝑓₁₂𝑞^𝑛𝑇_𝑞𝑔₁₂𝑇_𝑞𝐸₁₂,[𝐴,𝐵]≠0. (108) Therefore from (108), there exists a nonzero element 𝑧^−𝑛−1𝑓₁₂(𝑇_𝑞)𝐸₁₂ in (𝒮𝔖^±±_{𝑞,𝑘,𝑙})_{−𝑛−1/2}. Similarly, we prove that there exists nonzero 𝑧^−𝑛𝑓₂₁(𝑇_𝑞)𝐸₂₁ in (𝒮𝔖^±±_{𝑞,𝑘,𝑙})_{−𝑛−1/2}; finally the proof follows from (b). The proof of the converse is trivial.

Corollary 21. (a) 𝑎=𝑧⁻¹𝑓(𝑞^−1/2𝑇_𝑞)𝐸₁₂+𝑔(𝑇_𝑞)𝐸₂₁∈(𝒮𝔖^±±_{𝑞,𝑘,𝑙})_−1/2 is nondegenerate if and only if 𝑓 and 𝑔 are nonzero.
(b) 𝒮𝔖^±±_{𝑞,𝑘,𝑙} satisfies (SP3).

Proof. (a) Let 𝑎=𝑧⁻¹𝑓(𝑞^−1/2𝑇_𝑞)𝐸₁₂+𝑔(𝑇_𝑞)𝐸₂₁∈(𝒮𝔖^±±_{𝑞,𝑘,𝑙})_−1/2. Then, if 𝑓 and 𝑔 are nonzero by Lemma 20(b) and 𝔭^𝑎_−1/2 has finite codimension in (𝒮𝔖^±±_{𝑞,𝑘,𝑙})_−1/2, then by Lemma 20(c), 𝑎 is a nondegenerate element. Reciprocally, if 𝑓=0 or 𝑔=0, then by definition 𝔭^𝑎₋₁=0.
(b) Let 𝔭 be a nondegenerate parabolic subalgebra of 𝒮𝔖^±±_{𝑞,𝑘,𝑙}; then there exists 𝑎=𝑧⁻¹𝑓(𝑞⁻¹𝑇_𝑞)𝐸₁₂+𝑔(𝑇_𝑞)𝐸₂₁∈𝔭_−1/2 where 𝑓 and 𝑔 are nonzero. Then the proof follows from (a).

A functional 𝜆∈(𝒮𝔖^±±_{𝑞,𝑘,𝑙})^∗₀ is described by its labels, Δ_𝑛=𝜆(𝑇^𝑛_𝑞𝐸₁₁−𝑇_𝑞^−𝑛𝐸₂₂) with 𝑛∈ℤ and the central charge 𝜆(𝐶)=𝑐. Moreover, we define Δ⁰_𝑛=(Δ_𝑛−Δ_−𝑛)=𝜆((𝑇^𝑛_𝑞−𝑇_𝑞^−𝑛)𝐸₁₁+(𝑇^𝑛_𝑞−𝑇_𝑞^−𝑛)𝐸₂₂). We consider the generating series Δ_𝜆(𝑥)=_𝑛∈ℤΔ_𝑛𝑥^−𝑛,Δ⁰_𝜆(𝑥)=Δ_𝜆(𝑥)−Δ_𝜆𝑥⁻¹=_𝑛∈ℤΔ⁰_𝑛𝑥^−𝑛. (109)

Theorem 22. An irreducible highest weight 𝒮𝔖^±±_{𝑞,𝑘,𝑙}-module 𝐿(𝒮𝔖^±±_{𝑞,𝑘,𝑙},𝜆) is quasifinite if and only if one of the following equivalent conditions holds. (a)There exist Laurent polynomials 𝑓(𝑤) and 𝑔(𝑤) such that 𝑔(𝑥)Δ⁰_𝜆(𝑥)𝑓(𝑥)Δ=0,_0𝜆𝑞^−1/2𝑥=0. (110)(b)There exist quasipolynomials 𝑃₁₂(𝑥) and 𝑃₂₁(𝑥) such that 𝑃₁₂(𝑛)=𝑞^−𝑛/2Δ⁰_𝑛,𝑃₂₁(𝑛)=Δ⁰_𝑛, (111) for all 𝑛∈ℤ.

Proof. By Theorem 13, 𝐿(𝒮𝔖^±±_{𝑞,𝑘,𝑙},𝜆) is quasifinite if and only if there exists a nondegenerate element 𝑎 in (𝒮𝔖^±±_{𝑞,𝑘,𝑙})_−1/2, such that 𝜆([(𝒮𝔖^±±_{𝑞,𝑘,𝑙})_1/2,𝑎])=0.
Now, let 𝑎=𝑧⁻¹𝑓(𝑞^−1/2𝑇_𝑞)𝐸₁₂+𝑔(𝑇_𝑞)𝐸₂₁∈(𝒮𝔖^±±_{𝑞,𝑘,𝑙})_−1/2, with 𝑓(𝑤)=∑_𝑗𝑓_𝑗𝑤^𝑗∈ℂ[𝑤⁻¹,𝑤]^{(𝛿−1,𝑙−𝑘)} and 𝑔(𝑤)=∑_𝑗𝑔_𝑗𝑤^𝑗∈ℂ[𝑤⁻¹,𝑤]^{(𝛿′−1,−𝑙)} such that 𝑓 and 𝑔 are nonzero. Then by Corollary 21(a), 𝑎 is a nondegenerate element. Then, 𝜆([(𝒮𝔖_𝑞,𝑘)_1/2,𝑎])=0 if and only if, for all 𝑖∈ℤ, 𝜆𝑇^𝑖_𝑞−(−1)^𝛿0𝑇_𝑞^𝑙−𝑖𝐸₁₂𝜆𝑧𝑞,𝑎=0,_𝑖/2𝑇^𝑖_𝑞−(−1)^𝛿′0𝑞^{(𝑘−𝑙−𝑖)/2}𝑇_𝑞^{𝑘−𝑙−𝑖}𝐸₂₁,𝑎=0. (112)(113) From (112), we obtain that 0=𝜆𝑔𝑇_𝑞𝑇^𝑖_𝑞−(−1)^𝛿0𝑇_𝑞^𝑙−𝑖𝐼𝑑=_𝑗𝑔_𝑗Δ⁰_𝑖+𝑗. (114) Multiplying (114) by 𝑥^−𝑖 and adding over 𝑖∈ℤ, we obtain that 0=_𝑖,𝑗𝑔_𝑗Δ⁰_𝑖+𝑗𝑥^−𝑖=𝑔(𝑥)Δ⁰_𝜆(𝑥). (115) Then, by (113) and using Remark 8, we obtain that 0=𝜆_𝑗𝑓_𝑗𝑞^{(𝑖+𝑗)/2}𝑇_𝑞^𝑖+𝑗−_𝑗𝑓_𝑗𝑞^{−(𝑖+𝑗)/2}𝑇_𝑞^{−𝑖−𝑗}𝐸₁₁+_𝑗𝑓_𝑗𝑞^{−(𝑖+𝑗)/2}𝑇_𝑞^𝑖+𝑗−_𝑗𝑓_𝑗𝑞^{(𝑖+𝑗)/2}𝑇^{−𝑖−𝑗}_q𝐸₂₂+_𝑗𝑓_𝑗𝑞^{−(𝑖+𝑗)/2}𝑇_𝑞^𝑖+𝑗−_𝑗𝑓_𝑗𝑞^{(𝑖+𝑗)/2}𝑇_𝑞^{−𝑖−𝑗}₀𝑐=_𝑗𝑓_𝑗𝑞^{(𝑖+𝑗)/2}Δ_𝑖+𝑗−_𝑗𝑓_𝑗𝑞^{−(𝑖+𝑗)/2}Δ_{−𝑖−𝑗}. (116) Multiplying (116) by 𝑥^−𝑖 and adding over 𝑖∈ℤ, we obtain that 0=_𝑖,𝑗𝑓_𝑗𝑞^{(𝑖+𝑗)/2}Δ_𝑖+j𝑥^{−𝑖−𝑗}𝑥^𝑗−_𝑖,𝑗𝑓_𝑗𝑞^{−(𝑖+𝑗)/2}Δ_{−𝑖−𝑗}𝑥^{−𝑖−𝑗}𝑥^𝑗=_𝑗𝑓_𝑗Δ_𝜆𝑞^−1/2𝑥𝑥^𝑗−_𝑗𝑓_𝑗Δ_𝜆𝑞^1/2𝑥⁻¹𝑥^𝑗=𝑓(𝑥)Δ⁰_𝜆𝑞^−1/2𝑥. (117) 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).