Related papers: On intermediate Lie algebra $E_{7+1/2}$
A regular vertex operator algebra is a vertex operator algebra such that any weak module (without grading) is a direct sum of ordinary irreducible modules. In this paper we give several sufficient conditions under which a rational vertex…
The concept of a framed vertex operator algebra was studied in [DGH] (q-alg/9707008). This article is an analysis of all Virasoro frame stabilizers of the lattice VOA V for the E_8 root lattice, which is isomorphic to the E_8-level 1 affine…
We introduce and begin to study Lie theoretical analogs of symplectic reflection algebras for a finite cyclic group, which we call "cyclic double affine Lie algebra". We focus on type A : in the finite (resp. affine, double affine) case, we…
This book is mainly an exposition of the author's works and his joint works with his former students on explicit representations of finite-dimensional simple Lie algebras, related partial differential equations, linear orthogonal algebraic…
We study a vertex operator algebra containing a tensor product of Ising models. It is a direct sum of code vertex operator algebra and its irreducible modules. Therefore, we classify all irreducible modules of code vertex operator algebras…
We construct irreducible modules of centrally-extended classical Lie algebras over left ideals of the algebra of differential operators on the circle, through certain irreducible modules of centrally-extended classical Lie algebras of…
For an admissible affine vertex algebra $V_k(\mathfrak{g})$ of type $A$, we describe a new family of relaxed highest weight representations of $V_k(\mathfrak{g})$. They are simple quotients of representations of the affine Kac-Moody algebra…
In this paper, we determine all simple restricted modules over the mirror Heisenberg-Virasoro algebra ${\mathfrak{D}}$, and the twisted Heisenberg-Virasoro algebra $\bar\mathfrak{D}$ with nonzero level. As applications, we characterize…
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}…
The Witt algebra W_n is the Lie algebra of all derivations of the n-variable polynomial ring V_n=C[x_1, ..., x_n] (or of algebraic vector fields on A^n). A representation of W_n is polynomial if it arises as a subquotient of a sum of tensor…
The bc\beta\gamma-system W of rank 3 has an action of the affine vertex algebra V_0(sl_2), and the commutant vertex algebra C =Com(V_0(sl_2), W) contains copies of V_{-3/2}(sl_2) and Odake's algebra O. Odake's algebra is an extension of the…
The correspondence between four-dimensional $\mathcal{N}=2$ superconformal field theories and vertex operator algebras, when applied to theories of class $\mathcal{S}$, leads to a rich family of VOAs that have been given the monicker chiral…
We geometrically characterise the Veronese representations of ring projective planes over algebras which are analogues of the dual numbers, giving rise to projective Hjelmslev planes of level 2 coordinatised over quadratic alternative…
We study a special class of holomorphic vertex operator algebras (VOAs) that we call \emph{balanced}.\ For a balanced, holomorphic VOA $V=\mathbb{C}\mathbf{1}\oplus V_1\oplus\dots$ with $c=32$ or $40$ we show that the Virasoro vectors of…
By studying the properties of $q$-series $\widehat Z$-invariants, we develop a dictionary between 3-manifolds and vertex algebras. In particular, we generalize previously known entries in this dictionary to Lie groups of higher rank, to…
Certain vertex operator algebras have integral forms (integral spans of bases which are closed under the countable set of products). It is unclear when they (or integral multiples of them) are integral as lattices under the natural bilinear…
The Lie-algebraic method approximates differential operators that are formal polynomials of {1,x,d/dx} by linear operators acting on a finite dimensional space of polynomials. In this paper we prove that the rank of the n-dimensional…
Let $L_{D_8}(1, 0)$ and $L_{E_8}(1, 0)$ be the simple vertex operator algebras associated to untwisted affine Lie algebra $\widehat{{\mathbf g}}_{D_{8}}$ and $\widehat{{\mathbf g}}_{E_8}$ with level 1 respectively. In the 1980s by I.…
We discover a large class of simple affine vertex algebras $V_{k} (\mathfrak g)$, associated to basic Lie superalgebras $\mathfrak g$ at non-admissible collapsing levels $k$, having exactly one irreducible $\mathfrak g$-locally finite…
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…