scholarly journals VERTEX OPERATOR REPRESENTATIONS OF QUANTUM TORI AT ROOTS OF UNITY

2004 ◽  
Vol 06 (01) ◽  
pp. 195-220 ◽  
Author(s):  
Y. BILLIG ◽  
K. ZHAO
Keyword(s):  
Lie Algebra ◽  
Vertex Operator ◽  
Roots Of Unity ◽  
Index Set ◽  
Quantum Torus ◽  
Quantum Tori ◽  
Highest Weight ◽  
Level 1 ◽  
Outer Derivation ◽  
Operator Representations

As Lie algebra, we add the center c1 (and the outer derivation d1) to the quantum torus [Formula: see text] to give the extended torus Lie algebra [Formula: see text] (and [Formula: see text] respectively). Before the present paper, only some level 1 vertex operator representations for some [Formula: see text] (and [Formula: see text]) were constructed. In this paper, we first give vertex operator representations for [Formula: see text] where I is an arbitrary index set. By embedding some [Formula: see text] into [Formula: see text], we obtain a series of higher level vertex operator representations for [Formula: see text] and [Formula: see text]. Most of these vertex operator representations yield irreducible highest weight modules over these [Formula: see text]. Also their character formulas follow directly.

scholarly journals Combinatorial bases of modules for affine Lie algebra B 2(1)

2013 ◽  
Vol 11 (2) ◽  
Author(s):  
Mirko Primc
Keyword(s):  
Lie Algebra ◽  
Vertex Operator ◽  
Vertex Operator Algebra ◽  
Intertwining Operators ◽  
Type B ◽  
Highest Weight Module ◽  
Highest Weight ◽  
Affine Lie Algebra ◽  
Algebra Theory ◽  
Simple Currents

AbstractWe construct bases of standard (i.e. integrable highest weight) modules L(Λ) for affine Lie algebra of type B 2(1) consisting of semi-infinite monomials. The main technical ingredient is a construction of monomial bases for Feigin-Stoyanovsky type subspaces W(Λ) of L(Λ) by using simple currents and intertwining operators in vertex operator algebra theory. By coincidence W(kΛ0) for B 2(1) and the integrable highest weight module L(kΛ0) for A 1(1) have the same parametrization of combinatorial bases and the same presentation P/I.

scholarly journals A note on the zeroth products of Frenkel–Jing operators

2017 ◽  
Vol 16 (03) ◽  
pp. 1750053 ◽  
Author(s):  
Slaven Kožić
Keyword(s):  
Lie Algebra ◽  
Vertex Operator ◽  
Vertex Operator Algebra ◽  
Vertex Algebra ◽  
Infinite Dimensional ◽  
Enveloping Algebra ◽  
Highest Weight ◽  
Finite Dimensional ◽  
Quantum Vertex ◽  
Algebra Theory

Let [Formula: see text] be an untwisted affine Kac–Moody Lie algebra. The top of every irreducible highest weight integrable [Formula: see text]-module is the finite-dimensional irreducible [Formula: see text]-module, where the action of the simple Lie algebra [Formula: see text] is given by zeroth products arising from the underlying vertex operator algebra theory. Motivated by this fact, we consider zeroth products of level [Formula: see text] Frenkel–Jing operators corresponding to Drinfeld realization of the quantum affine algebra [Formula: see text]. By applying these products, which originate from the quantum vertex algebra theory developed by Li, on the extension of Koyama vertex operator [Formula: see text], we obtain an infinite-dimensional vector space [Formula: see text]. Next, we introduce an associative algebra [Formula: see text], a certain quantum analogue of the universal enveloping algebra [Formula: see text], and construct some infinite-dimensional [Formula: see text]-modules [Formula: see text] corresponding to the finite-dimensional irreducible [Formula: see text]-modules [Formula: see text]. We show that the space [Formula: see text] carries a structure of an [Formula: see text]-module and, furthermore, we prove that the [Formula: see text]-module [Formula: see text] is isomorphic to the [Formula: see text]-module [Formula: see text].

scholarly journals Uprolling unrolled quantum groups

2021 ◽  
pp. 2150023
Author(s):  
Thomas Creutzig ◽  
Matthew Rupert
Keyword(s):  
Quantum Group ◽  
Lie Algebra ◽  
Quantum Groups ◽  
Vertex Operator ◽  
Operator Algebras ◽  
Vertex Algebra ◽  
Vertex Operator Algebras ◽  
Roots Of Unity ◽  
Simple Lie Algebra ◽  
Higher Rank

We construct families of commutative (super) algebra objects in the category of weight modules for the unrolled restricted quantum group [Formula: see text] of a simple Lie algebra [Formula: see text] at roots of unity, and study their categories of local modules. We determine their simple modules and derive conditions for these categories being finite, non-degenerate, and ribbon. Motivated by numerous examples in the [Formula: see text] case, we expect some of these categories to compare nicely to categories of modules for vertex operator algebras. We focus in particular on examples expected to correspond to the higher rank triplet vertex algebra [Formula: see text] of Feigin and Tipunin and the [Formula: see text] algebras.

On the Classification of Rational Quantum Tori and the Structure of Their Automorphism Groups

2008 ◽  
Vol 51 (2) ◽  
pp. 261-282 ◽  
Author(s):  
Karl-Hermann Neeb
Keyword(s):  
Automorphism Group ◽  
Normal Form ◽  
Exact Sequence ◽  
Present Note ◽  
Automorphism Groups ◽  
Roots Of Unity ◽  
Quantum Torus ◽  
Twisted Group Algebra ◽  
Quantum Tori

AbstractAn n-dimensional quantum torus is a twisted group algebra of the group ℤn. It is called rational if all invertible commutators are roots of unity. In the present note we describe a normal form for rational n-dimensional quantum tori over any field. Moreover, we show that for n = 2 the natural exact sequence describing the automorphism group of the quantum torus splits over any field.

scholarly journals A RECURRENCE RELATION FOR CHARACTERS OF HIGHEST WEIGHT INTEGRABLE MODULES FOR AFFINE LIE ALGEBRAS

2007 ◽  
Vol 09 (02) ◽  
pp. 121-133 ◽  
Author(s):  
WILLIAM J. COOK ◽  
HAISHENG LI ◽  
KAILASH C. MISRA
Keyword(s):  
Recurrence Relation ◽  
Lie Algebras ◽  
Vertex Operator ◽  
Recurrence Relations ◽  
Operator Algebras ◽  
Affine Lie Algebras ◽  
Highest Weight ◽  
Irreducible Modules ◽  
Level 1 ◽  
Integrable Modules

Using certain results for the vertex operator algebras associated with affine Lie algebras, we obtain recurrence relations for the characters of integrable highest weight irreducible modules for an affine Lie algebra. As an application we show that in the simply-laced level 1 case, these recurrence relations give the known characters, whose principal specializations naturally give rise to some multisum Macdonald identities.

Integrable Representations for the Extended Affine Lie Algebra Coordinated by Quantum Tori in Two Variables

2014 ◽  
Vol 21 (03) ◽  
pp. 535-540 ◽  
Author(s):  
Fei Kong ◽  
Zhiqiang Li ◽  
Shaobin Tan ◽  
Qing Wang
Keyword(s):  
Lie Algebra ◽  
Type A ◽  
Quantum Torus ◽  
The Core ◽  
Quantum Tori ◽  
Finite Dimensional ◽  
Affine Lie Algebra ◽  
Extended Affine Lie Algebra ◽  
Integrable Modules

In this paper we classify the irreducible integrable modules for the core of the extended affine Lie algebra of type Ad-1 coordinated by ℂq with finite-dimensional weight spaces and the center acting trivially, where ℂq is the quantum torus in two variables.

scholarly journals IRREDUCIBLE HARISH CHANDRA MODULES OVER THE DERIVATION ALGEBRAS OF RATIONAL QUANTUM TORI

2013 ◽  
Vol 55 (3) ◽  
pp. 677-693 ◽  
Author(s):  
GENQIANG LIU ◽  
KAIMING ZHAO
Keyword(s):  
Positive Integer ◽  
Lie Algebra ◽  
Semidirect Product ◽  
Quantum Torus ◽  
Quantum Tori

AbstractLet d be a positive integer, q=(qij)d×d be a d×d matrix, ℂq be the quantum torus algebra associated with q. We have the semidirect product Lie algebra $\mathfrak{g}$=Der(ℂq)⋉Z(ℂq), where Z(ℂq) is the centre of the rational quantum torus algebra ℂq. In this paper, we construct a class of irreducible weight $\mathfrak{g}$-modules $\mathcal{V}$α (V,W) with three parameters: a vector α∈ℂd, an irreducible $\mathfrak{gl}$d-module V and a graded-irreducible $\mathfrak{gl}$N-module W. Then, we show that an irreducible Harish Chandra (uniformaly bounded) $\mathfrak{g}$-module M is isomorphic to $\mathcal{V}$α(V,W) for suitable α, V, W, if the action of Z(ℂq) on M is associative (respectively nonzero).

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