Representations of a class of lattice type vertex algebras

Berman, Stephen; Dong, Chongying; Tan, Shaobin

doi:10.1016/s0022-4049(02)00053-1

Cited by 33 publications

(28 citation statements)

References 14 publications

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“…The reason we consider only rank 2 quantum torus Lie algebras is the following. In many references like [2], [8], [9], [10] (but not [6]), higher rank quantum torus Lie algebras are studied but with the assumption that all the variables except t 1 are commutative. Algebras under this assumption essentially have the same properties which assure that they have the same type representations.…”

Section: ±1mentioning

confidence: 99%

“…For properties of C q , please refer to [2] or [12]. Define radf = {a ∈ Z Z 2 |f (a, Z Z 2 ) = 1} and…”

Section: ±1mentioning

confidence: 99%

“…Before this paper, there appeared only highest weight representations with finite dimensional weight spaces for C q and C q with level one or other positive integral levels (see [2], [8], [9], [5], [6]). In this paper, we define the highest weight irreducible (Z Z-graded) module V (φ) over C q and C q for any linear map φ : C[t …”

Section: ±1mentioning

confidence: 99%

“…We first establish the following Lemma. r > 1, a 1 , a 2 , ..., a r , β 1 , β 2 , ..., β r ∈ C with |β 1 | = |β 2 Proof. Suppose only one a k is not zero, say a 1 = 0.…”

Section: Thus We Have the Expression For φ(T Imentioning

confidence: 99%

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Highest weight irreducible representations of rank $2$ quantum tori

Rao

Zhao

2004

Mathematical Research Letters

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Abstract. For any nonzero q ∈ C (the complex numbers), the rank 2 quantum torus C q is the skew Laurent polynomial algebra2 ] with defining relations: t 2 t 1 = qt 1 t 2 and t i tHere we consider C q as the naturally associated Lie algebra. We add the one dimensional center Cc 1 and the outer derivation d 1 to C q to get the extended torus Lie algebra C q (and C q , in a different manner), where we assume q is a primitive m-th root of unity for C q . Before this paper, there appeared highest weight representations for C q and C q with only positive integral levels. In this paper, we define the highest weight irreducible (Z Z-graded) module V (φ) over C q and C q for any linear map φ : C[t ±12 ] + Cc 1 + Cd 1 → C, thus the central charge (level) can be any complex numbers. We obtain the necessary and sufficient conditions for V (φ) to have finite dimensional weight spaces, thus obtaining a lot of new irreducible weight representations for these Lie algebras. The corresponding irreducible Z Z × Z Z-graded modules with finite dimensional weight spaces over C q are also constructed.

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Section: ±1mentioning

confidence: 99%

“…For properties of C q , please refer to [2] or [12]. Define radf = {a ∈ Z Z 2 |f (a, Z Z 2 ) = 1} and…”

Section: ±1mentioning

confidence: 99%

Section: ±1mentioning

confidence: 99%

“…We first establish the following Lemma. r > 1, a 1 , a 2 , ..., a r , β 1 , β 2 , ..., β r ∈ C with |β 1 | = |β 2 Proof. Suppose only one a k is not zero, say a 1 = 0.…”

Section: Thus We Have the Expression For φ(T Imentioning

confidence: 99%

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Highest weight irreducible representations of rank $2$ quantum tori

Rao

Zhao

2004

Mathematical Research Letters

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“…For example, the vertex (operator) algebra associated with a βγ system, which plays a central role in free field realization of affine Lie algebras (see [Wakimoto 1986;Feȋgin and Frenkel 1988;1990a;1990b, Frenkel andBen-Zvi 2001]), is such an ‫-ގ‬graded vertex algebra. The vertex (operator) algebras constructed from toroidal Lie algebras are also of this type [Berman et al 2002a;2002b].…”

Section: Introductionmentioning

confidence: 99%

Twisted modules for vertex algebras associated with vertex algebroids

Li¹,

Yamskulna²

2007

Pacific J. Math.

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We continue the work in our earlier paper, "On certain vertex algebras and their modules associated with vertex algebroids", J. Algebra 283 (2005), 367-398, to construct and classify graded simple twisted modules for the ‫-ގ‬ graded vertex algebras constructed by Gorbounov, Malikov and Schechtman from vertex algebroids. In addition, we determine the full automorphism groups of those ‫-ގ‬graded vertex algebras in terms of the automorphism groups of the corresponding vertex algebroids.

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Free Field Realisation of the Chiral Universal Centraliser

Beem

Nair

2023

Ann. Henri Poincaré

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In the TQFT formalism of Moore–Tachikawa for describing Higgs branches of theories of class $${\mathcal {S}}$$ S , the space associated to the unpunctured sphere in type $${{\mathfrak {g}}}$$ g is the universal centraliser $${\mathfrak {Z}}_G$$ Z G , where $${{\mathfrak {g}}}=Lie(G)$$ g = L i e ( G ) . In more physical terms, this space arises as the Coulomb branch of pure $${\mathcal {N}}=4$$ N = 4 gauge theory in three dimensions with gauge group $${{\check{G}}}$$ G ˇ , the Langlands dual. In the analogous formalism for describing chiral algebras of class $${\mathcal {S}}$$ S , the vertex algebra associated to the sphere has been dubbed the chiral universal centraliser. In this paper, we construct an open, symplectic embedding from a cover of the Kostant–Toda lattice of type $${{\mathfrak {g}}}$$ g to the universal centraliser of G—extending a classic result of Kostant. Using this embedding and some observations on the Poisson algebraic structure of $${\mathfrak {Z}}_G$$ Z G , we propose a free field realisation of the chiral universal centraliser for any simple group G. We exploit this realisation to develop free field realisations of chiral algebras of class $${\mathcal {S}}$$ S of type $${{\mathfrak {a}}}_1$$ a 1 for theories of genus zero with $$n=1,\ldots ,6$$ n = 1 , … , 6 punctures. These realisations make generalised S-duality completely manifest, and the generalisation to $$n\geqslant 7$$ n ⩾ 7 punctures is conceptually clear, though technically burdensome.

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Representations of a class of lattice type vertex algebras

Cited by 33 publications

References 14 publications

Highest weight irreducible representations of rank $2$ quantum tori

Highest weight irreducible representations of rank $2$ quantum tori

Twisted modules for vertex algebras associated with vertex algebroids

Free Field Realisation of the Chiral Universal Centraliser

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