Related papers: Some developments in vertex operator algebra theor…
The radical of a vertex operator algebra associated to a module is defined and computed.
Motivated by the sharp contrast between classical and quantum physics as probability theories, in these lecture notes I introduce the basic notions of operator algebras that are relevant for the algebraic approach to quantum physics.…
This is an expository article about operads in homotopy theory written as a chapter for an upcoming book. It concentrates on what the author views as the basic topics in the homotopy theory of operadic algebras: the definition of operads,…
We consider the extension of the Heisenberg vertex operator algebra by all its irreducible modules. We give an elementary construction for the intertwining vertex operators and show that they satisfy a complex parametrized generalized…
This article will appear in the Encyclopedia of Mathematical Physics (Elsevier, 2006).
We present a vertex operator algebra which is an extension of the level $k$ vertex operator algebra for the $\hat{sl}_2$ conformal field theory. We construct monomial basis of its irreducible representations.
The quantum dimensions of modules for vertex operator algebras are defined and their properties are discussed. The possible values of the quantum dimensions are obtained for rational vertex operator algebras. A criterion for simple currents…
A higher dimensional analogue of the notion of vertex algebra is formulated in terms of formal variable language with Borcherds' notion of $G$-vertex algebra as a motivation. Some examples are given and certain analogous duality properties…
We prove the existence of algebras of hypercyclic vectors in three cases: convolution operators, composition operators, and backward shift operators.
We investigate a general structure theory for a vertex operator algebra. We discuss the center and blocks, the Jacobson radical and solvable radical and local vertex operator algebras. The main consequence of our structure theory is that if…
A key notion bridging the gap between {\it quantum operator algebras} \cite{LZ10} and {\it vertex operator algebras} \cite{Bor}\cite{FLM} is the definition of the commutativity of a pair of quantum operators (see section 2 below). This is…
We suggest a few projects for studying vertex algebras with emphasis on finite group viewpoints.
Differentiations of operator algebras over non-archimedean spherically complete fields are investigated. Theorems about a differentiation being internal are demonstrated.
Using recursion formulas for vertex operator algebra higher genus characters with formal parameters identified with local coordinates around marked points on a Riemann surface of arbitrary genus, we introduce the notion of a vertex operator…
In this paper we present the principal construction of the vertex operator representation for toroidal Lie algebras.
Vertex operators for the deformed Virasoro algebra are defined, their bosonic representation is constructed and difference equation for the simplest vertex operators is described.
We give a survey on the developments in a certain theory of quantum vertex algebras, including a conceptual construction of quantum vertex algebras and their modules and a connection of double Yangians and Zamolodchikov-Faddeev algebras…
These are the lecture notes for a course taught at Tsinghua University in the spring of 2022. In these notes, we develop the basic theory of vertex operator algebras (VOAs) and their conformal blocks using complex-analytic methods. In…
After an historical introduction on the standard algebraic approach to quantum mechanics of large systems we review the basic mathematical aspects of the algebras of unbounded operators. After that we discuss in some details their relevance…
We introduce two mirror constructions of Vertex Operator Algebras associated to special boundary conditions in 3d N=4 gauge theories. We conjecture various relations between these boundary VOA's and properties of the (topologically twisted)…