scholarly journals The Quantum Group Dual of the First-Row Subcategory for the Generic Virasoro VOA

2021 ◽  
Author(s):  
Shinji Koshida ◽  
Kalle Kytölä
Keyword(s):  
Quantum Group ◽  
Vertex Operator ◽  
Vertex Operator Algebra ◽  
Central Charge ◽  
Coulomb Gas ◽  
Intertwining Operators ◽  
Group Method ◽  
Conformal Blocks ◽  
Finite Dimensional

AbstractIn several examples it has been observed that a module category of a vertex operator algebra (VOA) is equivalent to a category of representations of some quantum group. The present article is concerned with developing such a duality in the case of the Virasoro VOA at generic central charge; arguably the most rudimentary of all VOAs, yet structurally complicated. We do not address the category of all modules of the generic Virasoro VOA, but we consider the infinitely many modules from the first row of the Kac table. Building on an explicit quantum group method of Coulomb gas integrals, we give a new proof of the fusion rules, we prove the analyticity of compositions of intertwining operators, and we show that the conformal blocks are fully determined by the quantum group method. Crucially, we prove the associativity of the intertwining operators among the first-row modules, and find that the associativity is governed by the 6j-symbols of the quantum group. Our results constitute a concrete duality between a VOA and a quantum group, and they will serve as the key tools to establish the equivalence of the first-row subcategory of modules of the generic Virasoro VOA and the category of (type-1) finite-dimensional representations of $${\mathcal {U}}_q (\mathfrak {sl}_2)$$ U q ( sl 2 ) .

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 BGG CATEGORY FOR THE QUANTUM SCHRÖDINGER ALGEBRA

2020 ◽  
pp. 1-14
Author(s):  
GENQIANG LIU ◽  
YANG LI
Keyword(s):  
Complex Number ◽  
Full Subcategory ◽  
Weyl Algebra ◽  
Central Charge ◽  
Enveloping Algebra ◽  
Root Of Unity ◽  
Finite Dimensional ◽  
Quantum Weyl Algebra ◽  
Schrödinger Algebra

Abstract In 1996, a q-deformation of the universal enveloping algebra of the Schrödinger Lie algebra was introduced in Dobrev et al. [J. Phys. A 29 (1996) 5909–5918.]. This algebra is called the quantum Schrödinger algebra. In this paper, we study the Bernstein-Gelfand-Gelfand (BGG) category $\mathcal{O}$ for the quantum Schrödinger algebra $U_q(\mathfrak{s})$ , where q is a nonzero complex number which is not a root of unity. If the central charge $\dot z\neq 0$ , using the module $B_{\dot z}$ over the quantum Weyl algebra $H_q$ , we show that there is an equivalence between the full subcategory $\mathcal{O}[\dot Z]$ consisting of modules with the central charge $\dot z$ and the BGG category $\mathcal{O}^{(\mathfrak{sl}_2)}$ for the quantum group $U_q(\mathfrak{sl}_2)$ . In the case that $\dot z = 0$ , we study the subcategory $\mathcal{A}$ consisting of finite dimensional $U_q(\mathfrak{s})$ -modules of type 1 with zero action of Z. We directly construct an equivalence functor from $\mathcal{A}$ to the category of finite dimensional representations of an infinite quiver with some quadratic relations. As a corollary, we show that the category of finite dimensional $U_q(\mathfrak{s})$ -modules is wild.

scholarly journals A quasi-Hopf algebra for the triplet vertex operator algebra

2019 ◽  
Vol 22 (03) ◽  
pp. 1950024 ◽  
Author(s):  
Thomas Creutzig ◽  
Azat M. Gainutdinov ◽  
Ingo Runkel
Keyword(s):  
Hopf Algebra ◽  
Quantum Group ◽  
Group Algebra ◽  
Vertex Operator ◽  
Operator Algebra ◽  
Vertex Operator Algebra ◽  
Root Of Unity

We give a new factorizable ribbon quasi-Hopf algebra [Formula: see text], whose underlying algebra is that of the restricted quantum group for [Formula: see text] at a [Formula: see text]th root of unity. The representation category of [Formula: see text] is conjecturally ribbon equivalent to that of the triplet vertex operator algebra (VOA) [Formula: see text]. We obtain [Formula: see text] via a simple current extension from the unrolled restricted quantum group at the same root of unity. The representation category of the unrolled quantum group is conjecturally equivalent to that of the singlet VOA [Formula: see text], and our construction is parallel to extending [Formula: see text] to [Formula: see text]. We illustrate the procedure in the simpler example of passing from the Hopf algebra for the group algebra [Formula: see text] to a quasi-Hopf algebra for [Formula: see text], which corresponds to passing from the Heisenberg VOA to a lattice extension.

scholarly journals GENUS-ZERO MODULAR FUNCTORS AND INTERTWINING OPERATOR ALGEBRAS

1998 ◽  
Vol 09 (07) ◽  
pp. 845-863 ◽  
Author(s):  
YI-ZHI HUANG
Keyword(s):  
Vertex Operator ◽  
Operator Algebra ◽  
Vertex Operator Algebra ◽  
Operator Algebras ◽  
Central Charge ◽  
Vertex Operator Algebras ◽  
Main Step ◽  
Group Representations ◽  
Genus Zero ◽  
Intertwining Operator

In [7] and [9], the author introduced the notion of intertwining operator algebra, a nonmeromorphic generalization of the notion of vertex operator algebra involving monodromies. The problem of constructing intertwining operator algebras from representations of suitable vertex operator algebras was solved implicitly earlier in [5]. In the present paper, we generalize the geometric and operadic formulation of the notion of vertex operator algebra given in [3, 4, 11, 12, 8] to the notion of intertwining operator algebra. We show that the category of intertwining operator algebras of central charge [Formula: see text] is isomorphic to the category of algebras over rational genus-zero modular functors (certain analytic partial operads) of central charge c satisfying a certain generalized meromorphicity property. This result is one main step in the construction of genus-zero conformal field theories from representations of vertex operator algebras announced in [7]. One byproduct of the proof of the present isomorphism theorem is a geometric construction of (framed) braid group representations from intertwining operator algebras and, in particular, from representations of suitable vertex operator algebras.

scholarly journals A LOGARITHMIC GENERALIZATION OF TENSOR PRODUCT THEORY FOR MODULES FOR A VERTEX OPERATOR ALGEBRA

2006 ◽  
Vol 17 (08) ◽  
pp. 975-1012 ◽  
Author(s):  
YI-ZHI HUANG ◽  
JAMES LEPOWSKY ◽  
LIN ZHANG
Keyword(s):  
Tensor Product ◽  
Vertex Operator ◽  
Operator Algebra ◽  
Vertex Operator Algebra ◽  
Vertex Algebra ◽  
Intertwining Operators

We describe a logarithmic tensor product theory for certain module categories for a "conformal vertex algebra". In this theory, which is a natural, although intricate, generalization of earlier work of Huang and Lepowsky, we do not require the module categories to be semisimple, and we accommodate modules with generalized weight spaces. The corresponding intertwining operators contain logarithms of the variables.

scholarly journals The automorphism group of the -orbifold of the Barnes–Wall lattice vertex operator algebra of central charge 32

2014 ◽  
Vol 156 (2) ◽  
pp. 343-361 ◽  
Author(s):  
HIROKI SHIMAKURA
Keyword(s):  
Automorphism Group ◽  
Vertex Operator ◽  
Operator Algebra ◽  
Vertex Operator Algebra ◽  
Central Charge ◽  
Group Structure ◽  
Graph Structure ◽  
Full Automorphism Group ◽  
Lattice Vertex Operator Algebra

AbstractIn this paper, we prove that the full automorphism group of the ${\mathbb Z}_2$-orbifold of the Barnes–Wall lattice vertex operator algebra of central charge 32 has the shape 227.E6(2). In order to identify the group structure, we introduce a graph structure on the Griess algebra and show that it is a rank 3 graph associated to E6(2).

scholarly journals On orbifold constructions associated with the Leech lattice vertex operator algebra

2018 ◽  
Vol 168 (2) ◽  
pp. 261-285 ◽  
Author(s):  
CHING HUNG LAM ◽  
HIROKI SHIMAKURA
Keyword(s):  
Lie Algebra ◽  
Vertex Operator ◽  
Operator Algebra ◽  
Vertex Operator Algebra ◽  
Operator Algebras ◽  
Central Charge ◽  
Leech Lattice ◽  
Weight One ◽  
Strongly Regular ◽  
Lattice Vertex Operator Algebra

AbstractIn this paper, we study orbifold constructions associated with the Leech lattice vertex operator algebra. As an application, we prove that the structure of a strongly regular holomorphic vertex operator algebra of central charge 24 is uniquely determined by its weight one Lie algebra if the Lie algebra has the type A3,43A1,2, A4,52, D4,12A2,6, A6,7, A7,4A1,13, D5,8A1,2 or D6,5A1,12 by using the reverse orbifold construction. Our result also provides alternative constructions of these vertex operator algebras (except for the case A6,7) from the Leech lattice vertex operator algebra.

REGULAR REPRESENTATIONS OF VERTEX OPERATOR ALGEBRAS

2002 ◽  
Vol 04 (04) ◽  
pp. 639-683 ◽  
Author(s):  
HAISHENG LI
Keyword(s):  
Tensor Product ◽  
Vertex Operator ◽  
Dual Space ◽  
Vertex Operator Algebra ◽  
Operator Algebras ◽  
Canonical Decomposition ◽  
Vertex Operator Algebras ◽  
Intertwining Operators ◽  
Linear Isomorphism ◽  
Regular Representations

This paper is to establish a theory of regular representations for vertex operator algebras. In the paper, for a vertex operator algebra V and a V-module W, we construct, out of the dual space W*, a family of canonical weak V ⊗ V-modules [Formula: see text] with a nonzero complex number z as the parameter. We prove that for V-modules W, W1 and W2, a P(z)-intertwining map of type [Formula: see text] in the sense of Huang and Lepowsky exactly amounts to a V ⊗ V-homomorphism from W1 ⊗ W2 into [Formula: see text]. Combining this with Huang and Lepowsky's one-to-one linear correspondence between the space of intertwining operators and the space of P(z)-intertwining maps of the same type we obtain a canonical linear isomorphism fromthe space [Formula: see text] of intertwining operators of the indicated type to [Formula: see text]. Denote by RP(z)(W) the sum of all (ordinary) V ⊗ V-submodules of [Formula: see text]. Assuming that V satisfies certain suitable conditions, we obtain a canonical decomposition of RP(z)(W) into irreducible V ⊗ V-modules. In particular, we obtain a decomposition of Peter–Weyl type for RP(z)(V). Denote by ℱP(z) the functor from the category of V-modules to the category of weak V ⊗ V-modules such that ℱP(z)(W)=RP(z)(W'). We prove that for V-modules W1, W2, a P(z)-tensor product of W1 and W2 in the sense of Huang and Lepowsky exactly amounts to a universal from W1 ⊗ W2 to the functor ℱP(z). This implies that the functor ℱP(z) is essentially a right adjoint of the Huang–Lepowsky's P(z)-tensor product functor. It is also proved that RP(z)(W) for [Formula: see text] are canonically isomorphic V ⊗ V-modules.

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