Extension of Vertex Operator Algebra $V_{\widehat{H}_{4}}(\ell,0)$

2014 ◽  
Vol 21 (03) ◽  
pp. 361-380 ◽  
Author(s):  
Cuipo Jiang ◽  
Song Wang
Keyword(s):  
Representation Theory ◽  
Lie Algebra ◽  
Vertex Operator ◽  
Operator Algebra ◽  
Vertex Operator Algebra ◽  
Vertex Algebra ◽  
Irreducible Representations ◽  
Natural Conditions ◽  
Irreducible Modules ◽  
Heisenberg Algebras

We classify the irreducible restricted modules for the affine Nappi-Witten Lie algebra [Formula: see text] with some natural conditions. It turns out that the representation theory of [Formula: see text] is quite different from the theory of representations of Heisenberg algebras. We also study the extension of the vertex operator algebra [Formula: see text] by the even lattice L. We give the structure of the extension [Formula: see text] and its irreducible modules via irreducible representations of [Formula: see text] viewed as a vertex algebra.

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 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 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.

scholarly journals SEMI-INFINITE FORMS AND TOPOLOGICAL VERTEX OPERATOR ALGEBRAS

2000 ◽  
Vol 02 (02) ◽  
pp. 191-241 ◽  
Author(s):  
YI-ZHI HUANG ◽  
WENHUA ZHAO
Keyword(s):  
Moduli Spaces ◽  
Vertex Operator ◽  
Operator Algebra ◽  
Riemann Surfaces ◽  
Vertex Operator Algebra ◽  
Operator Algebras ◽  
Vertex Algebra ◽  
Vertex Operator Algebras ◽  
Topological Vertex ◽  
Type K

Semi-infinite forms on the moduli spaces of genus-zero Riemann surfaces with punctures and local coordinates are introduced. A partial operad of semi-infinite forms is constructed. Using semi-infinite forms and motivated by a partial suboperad of the partial operad of semi-infinite forms, topological vertex partial operads of type k<0 and strong topological vertex partial operads of type k<0 are constructed. It is proved that the category of (locally-)grading-restricted (strong) topological vertex operator algebras of type k<0 and the category of (weakly) meromorphic ℤ×ℤ-graded algebras over the (strong) topological vertex partial operad of type k are isomorphic. As an application of this isomorphism theorem, the following conjecture of Lian-Zuckerman and Kimura-Voronov-Zuckerman is proved: A strong topological vertex operator algebra gives a (weak) homotopy Gerstenhaber algebra. These results hold in particular for the tensor product of the moonshine module vertex operator algebra, the vertex algebra constructed from a rank 2 Lorentz lattice and the ghost vertex operator algebra, studied in detail first by Lian and Zuckerman.

scholarly journals Vertex Algebras Associated with Hypertoric Varieties

2020 ◽  
Author(s):  
Toshiro Kuwabara
Keyword(s):  
Vertex Operator ◽  
Operator Algebra ◽  
Vertex Operator Algebra ◽  
Vertex Algebra ◽  
Hamiltonian Reduction ◽  
Vertex Algebras ◽  
Torus Action ◽  
Level 1 ◽  
Hypertoric Varieties ◽  
Construction Works

Abstract We construct a family of vertex algebras associated with a family of symplectic singularity/resolution, called hypertoric varieties. While the hypertoric varieties are constructed by a certain Hamiltonian reduction associated with a torus action, our vertex algebras are constructed by (semi-infinite) BRST reduction. The construction works algebro-geometrically, and we construct sheaves of $\hbar $-adic vertex algebras over hypertoric varieties, which localize the vertex algebras. We determine when it is a vertex operator algebra by giving an explicit conformal vector. We also discuss the Zhu algebra of the vertex algebra and its relation with a quantization of the hypertoric variety. In certain cases, we obtain the affine ${\mathcal{W}}$-algebra associated with the subregular nilpotent orbit in $\mathfrak{s}\mathfrak{l}_N$ at level $N-1$ and simple affine vertex operator algebra for $\mathfrak{s}\mathfrak{l}_N$ at level $-1$.

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