Twisted Vertex Operators and Unitary Lie Algebras

Chen, Fulin; Gao, Yun; Jing, Naihuan; Tan, Shaobin

doi:10.4153/cjm-2014-010-1

Cited by 4 publications

(4 citation statements)

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“…This section is devoted to the application of Theorem 5.2. By choosing some special quadruples (Q, ν, m, Γ), we recover vertex operator representations presented in [L,G1,G2,BS,BGT,CGJT,CT]. We also provide a vertex operator representation for the BC N −1 -graded Lie algebra o (2) 2N (C Γ ).…”

Section: Applicationsmentioning

confidence: 99%

Twisted Γ-Lie algebras and their vertex operator representations

2015

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Let Γ be a generic subgroup of the multiplicative group C * of nonzero complex numbers. We define a class of Lie algebras associated to Γ, called twisted Γ-Lie algebras, which is a natural generalization of the twisted affine Lie algebras. Starting from an arbitrary even sublattice Q of Z N and an arbitrary finite order isometry of Z N preserving Q, we construct a family of twisted Γ-vertex operators acting on generalized Fock spaces which afford irreducible representations for certain twisted Γ-Lie algebras. As application, this recovers a number of known vertex operator realizations for infinite dimensional Lie algebras, such as twisted affine Lie algebras, extended affine Lie algebras of type A, trigonometric Lie algebras of series A and B, unitary Lie algebras, and BC-graded Lie algebras. 1 (A2). Q is a sublattice of P such that α, α ∈ 2Z for α ∈ Q. (A3). ν is an isometry of P preserving Q. That is, ν is an automorphism of P such that να, νβ = α, β for α, β ∈ P and ν(Q) = Q.(A4). m is a positive integer such that ν m = Id, the identity map on P .Note that if the triple (Q, ν, m) satisfies the assumptions (A2)-(A4), then the assumption (A5) can always be arranged by doubling m if necessary. Now we give some examples of the triples (Q, ν, m) which satisfy assumptions (A2)-(A5). Let Q = Q(A N −1 ), N ≥ 2 or Q(D N ), N ≥ 5 be the root lattices of type A N −1 and D N respectively, and let ν be an isometry of Q. It is known that ν can be lifted to be an isometry of P and has finite order. Thus there exists a positive integer m such that (Q, ν, m) satisfies assumptions (A2)-(A5).Starting from a quadruple (Q, ν, m, Γ) which satisfies (A1)-(A5), we construct a twisted Γ-Lie algebra G(Q, ν, m, Γ) and define a family of twisted Γ-vertex operators acting on a generalized Fock space. By computing the commutator relations of these twisted Γ-vertex operators, we obtain a class of irreducible representations for the twisted Γ-Lie algebra G(Q, ν, m, Γ). One will see that it is very subtle and technical to determine the commutator relations for the twisted Γ-vertex operators. To do this, we develop a highly non-trivial generalization of the combinatorial identity presented in Proposition 4.1 of [L]. As applications, we show that with different choices of quadruples (Q, ν, m, Γ), we recover the vertex operator constructions of the twisted affine Lie algebras, extended affine Lie algebras of type A (both homogeneous and principal constructions), trigonometric Lie algebras of series A and B, unitary Lie algebras and BC-graded Lie algebras given in [L, G-KL1, G-KL2, BS, G1, G2, BGT, CGJT, CT] respectively. Moreover, we also present vertex operator representations for a new twisted Γ-Lie algebras.The paper is organized as follows. In Section 2 we give the definition of the twisted Γ-Lie algebra G(Q, ν, m, Γ) for any quadruple (Q, ν, m, Γ) satisfying the assumptions (A1)-(A5). In Section 3 we define the generalized Fock space and give the twisted Γ-vertex operators. In Section 4 we establish a crucial identity (see (4.10)) by using ...

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Section: Applicationsmentioning

confidence: 99%

Twisted Γ-Lie algebras and their vertex operator representations

2015

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“…Recently, Gao and Jing [5] introduced a quantum Tits-Kantor-Kocher (TKK) algebra using homogeneous q-deformed vertex operators and the construction can be viewed as a generalization of the TKK algebra as a special unitary Lie algebra [4]. More recently, the general homogeneous construction has been generalized to the twisted setting [10]. This paper proposes a twisted version of the quantum TKK algebra as a twisted quantum toroidal algebra of type A 1 and also constructs its Fock space realization using the "−1" twisted vertex operators.…”

Section: Introductionmentioning

confidence: 99%