Related papers: Level-Rank Duality for Vertex Operator Algebras of…

For the semisimple Lie algebra $ \frak{sl}_n$, the basic representation $L_{\widehat{\frak{sl}_{n}}}(1,0)$ of the affine Lie algebra $\widehat{\frak{sl}_{n}}$ is a lattice vertex operator algebra. The first main result of the paper is to…

For the Klein group $K$, $k\in\mathbb{Z}_{\geqslant 1}$ and $m\in\mathbb{Z}_{\geqslant 4}$, we study the representations of the orbifold vertex operator algebra $L_{\hat{\mathfrak{sl}_2}}(k,0)^{K}$ and the commutant vertex operator algebra…

Let $L(-{1/2}(l+1),0)$ be the simple vertex operator algebra associated to an affine Lie algebra of type $A_{l}^{(1)}$ with the lowest admissible half-integer level $-{1/2}(l+1)$, for even l. We study the category of weak modules for that…

Let L(n-l+1/2,0) be the vertex operator algebra associated to an affine Lie algebra of type B_l^(1) at level n-l+1/2, for a positive integer n. We classify irreducible L(n-l+1/2,0)-modules and show that every L(n-l+1/2,0)-module is…

Let $L_{D_{\ell}}(-\ell +{3/2},0)$ (resp. $L_{B_{\ell}}(-\ell +{3/2},0)$) be the simple vertex operator algebra associated to affine Lie algebra of type $D_{\ell}^{(1)}$ (resp. $B_{\ell}^{(1)}$) with the lowest admissible half-integer level…

We give a coset realization of the vertex operator algebra $M(1)^+$ with central charge $\ell$. We realize $M(1)^+$ as a commutant of certain affine vertex algebras of level -1 in the vertex algebra $L_{C_{\ell}…

Here we initiate an investigation of the equational classes of m-symmetric algebras endowed with two tense operators. These varieties is a generalization of tense algebras. Our main interest is the duality theory for these classes of…

Duality relations between Lie algebras are a significant phenomenon in Lie algebra representation theory, with level-rank duality as a famous example. Level-rank dualities for affine Lie algebras of type $A^{(1)}$ were first discovered by…

Let Vect(R) be the Lie algebra of smooth vector fields on R. The space of symbols Pol(T^* R) admits a non-trivial deformation (given by differential operators on weighted densities) as a Vect(R)-module that becomes trivial once the action…

This is a continuation of our work to understand vertex operator algebras using the geometric properties of varieties attached to vertex operator algebras. For a class of vertex operator algebras including affine vertex operator algebras…

A Lax operator algebra is constructed for an arbitrary semi-simple Lie algebra over $\mathbb C$ equipped with a $\mathbb Z$-grading, and arbitrary compact Riemann surface with marked points. In this set-up, a treatment of almost graded…

This paper is to study vertex operator superalgebras which are strongly generated by their weight-$2$ and weight-$\frac{3}{2}$ homogeneous subspaces. Among the main results, it is proved that if such a vertex operator superalgebra $V$ is…

We continue the study of the vertex operator algebra $L(k,0)$ associated to a type $G_2^{(1)}$ affine Lie algebra at admissible one-third integer levels, $k = -2 + m + \tfrac{i}{3}\ (m\in \mathbb{Z}_{\ge 0}, i = 1,2)$, initiated in…

In this paper, we study representations of the vertex operator algebra $L(k,0)$ at one-third admissible levels $k= -5/3, -4/3, -2/3$ for the affine algebra of type $G_2^{(1)}$. We first determine singular vectors and then obtain a…

We notice that for any positive integer $k$, the set of $(1,2)$-specialized characters of level $k$ standard $A_{1}^{(1)}$-modules is the same as the set of rescaled graded dimensions of the subspaces of level $2k+1$ standard…

Classical invariants for representations of one Lie group can often be related to invariants of some other Lie group. Physics suggests that the right objects to consider for these questions are certain refinements of classical invariants…

A double algebra is a linear space $V$ equipped with linear map $V\otimes V\to V\otimes V$. Additional conditions on this map lead to the notions of Lie and associative double algebras. We prove that simple finite-dimensional Lie double…

In this paper we study a series of vertex operator algebras of integer level associated to the affine Lie algebra $A_{\ell}^{(1)}$. These vertex operator algebras are constructed by using the explicit construction of certain singular…

A vector space S of linear operators between vector spaces U and V is called locally linearly dependent (in abbreviated form: LLD) when every vector x of U is annihilated by a non-zero operator in S. A duality argument bridges the theory of…

Let $G$ be a simple complex Lie group with Lie algebra $\mf g$ and let $\af$ be the affine Lie algebra. We use intertwining operators and Knizhnik-Zamolodchikov equations to construct a family of $\N$-graded vertex operator algebras…