Searcharxiv⌕ Search

arXiv subjects

Haisheng Li

Publications and source records attributed to Haisheng Li.

At least 91 records · Page 5Linked to original sources
arXiv2002

Axiomatic $G_{1}$-vertex algebras

Inspired by the Borcherds' work on ``$G$-vertex algebras,'' we formulate and study an axiomatic counterpart of Borcherds' notion of $G$-vertex algebra for the simplest nontrivial elementary vertex group, which we denote by $G_{1}$. Specifically, we formulate a notion of axiomatic $G_{1}$-vertex algebra, prove certain basic properties and give certain examples, where the notion of axiomatic $G_{1}$-vertex algebra is a nonlocal generalization of the notion of vertex algebra. We also show how to construct axiomatic $G_{1}$-vertex algebras from a set of compatible $G_{1}$-vertex operators. The results of this paper were reported in June 2001, at the International Conference on Lie Algebras in the Morningside center, Beijing, China, and were reported on November 30, 2001, in the Quantum Mathematics Seminar, at Rutgers-New Brunswick. We noticed that a paper of Bakalov and Kac appeared today (math.QA/0204282) on noncommutative generalizations of vertex algebras, which has certain overlaps with the current paper. On the other hand, most of their results are orthogonal to the results of this paper.

math.QA↗
arXiv2001

On certain higher dimensional analogues of vertex algebras

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 are proved. Furthermore, it is proved that for any vector space $W$, any set of mutually local multi-variable vertex operators on $W$ in a certain canonical way generates a vertex algebra with $W$ as a natural module.

math.QA↗
arXiv2001

Regular representations and Huang-Lepowsky's tensor functors for vertex operator algebras

This is the second paper in a series to study regular representations for vertex operator algebras. In this paper, given a module $W$ for a vertex operator algebra $V$, we construct, out of the dual space $W^{*}$, a family of canonical (weak) $V\otimes V$-modules called ${\cal{D}}_{Q(z)}(W)$ parametrized by a nonzero complex number $z$. We prove that for $V$-modules $W,W_{1}$ and $W_{2}$, a $Q(z)$-intertwining map of type ${W'\choose W_{1}W_{2}}$ in the sense of Huang and Lepowsky exactly amounts to a $V\otimes V$-homomorphism from $W_{1}\otimes W_{2}$ to ${\cal{D}}_{Q(z)}(W)$ and that a $Q(z)$-tensor product of $V$-modules $W_{1}$ and $W_{2}$ in the sense of Huang and Lepowsky amounts to a universal from $W_{1}\otimes W_{2}$ to the functor ${\cal{F}}_{Q(z)}$, where ${\cal{F}}_{Q(z)}$ is a functor from the category of $V$-modules to the category of weak $V\otimes V$-modules defined by ${\cal{F}}_{Q(z)}(W)={\cal{D}}_{Q(z)}(W')$ for a $V$-module $W$. Furthermore, Huang-Lepowsky's $P(z)$ and $Q(z)$-tensor functors for the category of $V$-modules are extended to functors $T_{P(z)}$ and $T_{Q(z)}$ from the category of $V\otimes V$-modules to the category of $V$-modules. It is proved that functors ${\cal{F}}_{P(z)}$ and ${\cal{F}}_{Q(z)}$ are right adjoints of $T_{P(z)}$ and $T_{Q(z)}$, respectively.

math.QA↗
arXiv2000

On abelian generalized vertex algebras

This paper studies the algebraic aspect of a general abelian coset theory with a work of Dong and Lepowsky as our main motivation. It is proved that the vacuum space $Ω_{V}$ (or the space of highest weight vectors) of a Heisenberg algebra in a general vertex operator algebra $V$ has a natural generalized vertex algebra structure in the sense of Dong and Lepowsky and that the vacuum space $Ω_{W}$ of a $V$-module $W$ is a natural $Ω_{V}$-module. The automorphism group $\Aut_{Ω_{V}}Ω_{V}$ of the adjoint $Ω_{V}$-module is studied and it is proved to be a central extension of a certain torsion free abelian group by $\C^{\times}$. For certain subgroups $A$ of $\Aut_{Ω_{V}}Ω_{V}$, certain quotient algebras $Ω_{V}^{A}$ of $Ω_{V}$ are constructed. Furthermore, certain functors among the category of $V$-modules, the category of $Ω_{V}$-modules and the category of $Ω_{V}^{A}$-modules are constructed and irreducible $Ω_{V}$-modules and $Ω_{V}^{A}$-modules are classified in terms of irreducible $V$-modules. If the category of $V$-modules is semisimple, then it is proved that the category of $Ω_{V}^{A}$-modules is semisimple.

math.QA↗
arXiv2000

Generalized vertex algebras generated by parafermion-like vertex operators

It is proved that for a vector space W, any set of parafermion-like vertex operators on W in a certain canonical way generates a generalized vertex algebra in the sense of [DL2] with W as a natural module. This result generalizes a result of [Li2]. As an application, generalized vertex algebras are constructed from Lepowsky-Wilson's Z-algebras of any nonzero level.

math.QA↗
arXiv2000

The regular representations and the $A_{n}(V)$ algebras

For a vertex operator algebra $V$, the regular representations are related to the $A_{n}(V)$-algebras and their bimodules, and induced $V$-modules from $A_{n}(V)$-modules are defined and studied in terms of the regular representations.

math.QA↗
arXiv2000

Certain extensions of vertex operator algebras of affine type

We generalize Feigin and Miwa's construction of extended vertex operator (super)algebras $A_{k}(sl(2))$ for other types of simple Lie algebras. For all the constructed extended vertex operator (super)algebras, irreducible modules are classified, complete reducibility of every module is proved and fusion rules are determined modulo the fusion rules for vertex operator algebras of affine type.

math.QA↗
arXiv1999

Regular representations of vertex operator algebras, I

In this paper, given a module $W$ for a vertex operator algebra $V$ and a nonzero complex number $z$ we construct a canonical (weak) $V\otimes V$-module ${\cal{D}}_{P(z)}(W)$ (a subspace of $W^{*}$ depending on $z$). We prove that for $V$-modules $W, W_{1}$ and $W_{2}$, a $P(z)$-intertwining map of type ${W'\choose W_{1}W_{2}}$ ([H3], [HL0-3]) exactly amounts to a $V\otimes V$-homomorphism from $W_{1}\otimes W_{2}$ into ${\cal{D}}_{P(z)}(W)$. Using 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 from the space ${\cal{V}}^{W'}_{W_{1}W_{2}}$ of intertwining operators of the indicated type to $\Hom_{V\otimes V}(W_{1}\otimes W_{2},{\cal{D}}_{P(z)}(W))$. In the case that $W=V$, we obtain a decomposition of Peter-Weyl type for ${\cal{D}}_{P(z)}(V)$, which are what we call the regular representations of $V$.

math.QA↗
arXiv1999

On Z-graded associative algebras and their N-graded modules

Let $A$ be a $Z$-graded associative algebra and let $ρ$ be an irreducible $N$-graded representation of $A$ on $W$ with finite-dimensional homogeneous subspaces. Then it is proved that $ρ(\tilde{A})=gl_{J}(W)$, where $\tilde{A}$ is the completion of $A$ with respect to a certain topology and $gl_{J}(W)$ is the subalgebra of $\End W$, generated by homogeneous endomorphisms. It is also proved that an $N$-graded vector space $W$ with finite-dimensional homogeneous spaces is the only continuous irreducible $N$-graded $gl_{J}(W)$-module up to equivalence, where $gl_{J}(W)$ is considered as a topological algebra in a certain natural way, and that any continuous $N$-graded $gl_{J}(W)$-module is a direct sum of some copies of $W$. A duality for certain subalgebras of $gl_{J}(W)$ is also obtained.

math.QA↗
arXiv1998

Certain generating subspaces for vertex operator algebras

We study minimal generating subspaces of "weak PBW type" for vertex operator algebras that satisfy a certain condition on the grading, and we develop a procedure for finding such subspaces. As applications, we derive some results on generalized modules and produce a minimal generating space of weak PBW type for the vertex operator algebra $V_L$ associated with any positive-definite even lattice $L$.

math.QA↗
arXiv1998

Some finiteness properties of regular vertex operator algebras

We give a natural extension of the notion of the contragredient module for a vertex operator algebra. By using this extension we prove that for regular vertex operator algebras, Zhu's $C_{2}$-finiteness condition holds, fusion rules are finite and the vertex operator algebras themselves are finitely generated.

math.QA↗
arXiv1997

Modular invariance of trace functions in orbifold theory

The goal of the present paper is to provide a mathematically rigorous foundation to certain aspects of rational orbifold conformal field theory, in other words the theory of rational vertex operator algebras and their automorphisms. Under a certain finiteness condition on a rational vertex operator algebra V which holds in all known examples, we determine the precise numbers of g-twisted sectors for any automorphism g of V of finite order. We prove that the trace functions and correlations functions associated with such twisted sectors are holomorphic functions in the upper half-plane and, under suitable conditions, afford a representations of the modular group of the type prescribed in string theory. We establish the rationality of conformal weights and central charge. In addition to conformal field theory itself, where our conclusions are required on physical grounds, there are applications to the generalized Moonshine conjectures of Conway-Norton-Queen and to equivariant elliptic cohomology.

q-alg↗
arXiv1997

Vertex operator algebras and associative algebras

Let V be a vertex operator algebra. We construct a sequence of associative algebras A_n(V) (n=0,1,2,...) such that A_{n}(V) is a quotient of A_{n+1}(V) and a pair of functors between the category of A_n(V)-modules which are not A_{n-1}(V)-modules and the category of admissible V-modules. These functors exhibit a bijection between the simple modules in each category. We also show that V is rational if and only if all A_n(V) are finite-dimensional semisimple algebras.

q-alg↗
arXiv1996

Compact automorphism groups of vertex operator algebras

Let $V$ be a simple vertex operator algebra which admits the continuous, faithful action of a compact Lie group $G$ of automorphisms. We establish a Schur-Weyl type duality between the unitary, irreducible modules for $G$ and the irreducible modules for $V^G$ which are contained in $V$ where $V^G$ is the space of $G$-invariants of $V.$ We also prove a concomitant Galois correspondence between vertex operator subalgebras of $V$ which contain $V^G$ and closed Lie subgroups of $G$ in the case that $G$ is abelian.

q-alg↗
arXiv1996

Certain associative algebras similar to $U(sl_{2})$ and Zhu's algebra $A(V_{L})$

It is proved that Zhu's algebra for vertex operator algebra associated to a positive-definite even lattice of rank one is a finite-dimensional semiprimitive quotient algebra of certain associative algebra introduced by Smith. Zhu's algebra for vertex operator algebra associated to any positive-definite even lattice is also calculated and is related to a generalization of Smith's algebra.

q-alg↗