Toward a vertex operator construction of quantum affine algebras

We describe a construction of the quantum-deformed affine algebras using vertex operators in the free field theory. We prove the Serre relations for the Borel subalgebras of quantum affine algebras; in particular, we consider the case in detail. We also construct the generators corresponding to the positive roots of . __________ Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 154, No. 2, pp. 240–248, February, 2008.     

Representations of a class of lattice type vertex algebras

In this paper we study the representation theory for certain “half lattice vertex algebras.” In particular we construct a large class of irreducible modules for these vertex algebras. We also discuss how the representation theory of these vertex algebras are related to the representation theory of some associative algebras.     

Parafermion vertex operator algebras

This paper reviews some recent results on the parafermion vertex operator algebra associated to the integrable highest weight module L(k, 0) of positive integer level k for any affine Kac-Moody Lie algebra ĝ, where g is a finite dimensional simple Lie algebra. In particular, the generators and the C ₂-cofiniteness of the parafermion vertex operator algebras are discussed. A proof of the well-known fact that the parafermion vertex operator algebra can be realized as the commutant of a lattice vertex operator algebra in L(k, 0) is also given.     

Bimodules associated to vertex operator algebras

Let V be a vertex operator algebra and m, n ≥ 0. We construct an A _n (V)-A _m (V)-bimodule A _n,m (V) which determines the action of V from the level m subspace to level n subspace of an admissible V-module. We show how to use A _n,m (V) to construct naturally admissible V-modules from A _m (V)-modules. We also determine the structure of A _n,m (V) when V is rational. Chongying Dong was supported by NSF grants, China NSF grant 10328102 and a Faculty research grant from the University of California at Santa Cruz. Cuipo Jiang was supported in part by China NSF grant 10571119.     

Construction of vertex operator algebras from commutative associative algebras

Given a commutative associative algebra A with an associative form (’), we construct a vertex operator algebra V with the weight two space V₂;? A If in addition the form (’) is nondegenerate, we show that there is a simple vertex operator algebra with V₂;? A We also show that if A is semisimple, then the vertex operator algebra constructed is the tensor products of a certain number of Virasoro vertex operator algebras.     

The varieties of Heisenberg vertex operator algebras

For a vertex operator algebra V with conformal vector ω,we consider a class of vertex operator subalgebras and their conformal vectors.They are called semi-conformal vertex operator subalgebras and semiconformal vectors of(V,ω),respectively,and were used to study duality theory of vertex operator algebras via coset constructions.Using these objects attached to(V,ω),we shall understand the structure of the vertex operator algebra(V,ω).At first,we define the set Sc(V,ω)of semi-conformal vectors of V, then we prove that Sc(V,ω)is an affine algebraic variety with a partial ordering and an involution map.Corresponding to each semi-conformal vector,there is a unique maximal semi-conformal vertex operator subalgebra containing it.The properties of these subalgebras are invariants of vertex operator algebras.As an example,we describe the corresponding varieties of semi-conformal vectors for Heisenberg vertex operator algebras.As an application,we give two characterizations of Heisenberg vertex operator algebras using the properties of these varieties.     

Vertex operator algebras and representations of affine Lie algebras

We announce the construction of an explicit basis for all integrable highest weight modules over the Lie algebra A ₁ ⁽¹⁾. The construction uses representations of vertex operator algebras and leads to combinatorial identities of Rogers-Ramanujan-type.     

Representations as elements in affine composition algebras

Let be the path algebra of a Euclidean quiver over a finite field . The aim of this paper is to classify the modules with the property , where is Ringel's composition algebra. Namely, the main result says that if , then if and only if the regular direct summand of is a direct sum of modules from non-homogeneous tubes with quasi-dimension vectors non-sincere. The main methods are representation theory of affine quivers, the structure of triangular decompositions of tame composition algebras, and the invariant subspaces of skew derivations. As an application, we see that if and only if the quiver of is of Dynkin type.     

Isometries of certain operator algebras

To a given basis on an -dimensional Hilbert space , we associate the algebra of all linear operators on having every as an eigenvector. So, is commutative, semisimple, and -dimensional. Given two algebras of this type, and , there is a natural algebraic isomorphism of and . We study the question: When does preserve the operator norm?

     

Representations of centrally-extended Lie algebras over differential operators and vertex 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 infinite matrices with finite number of nonzero entries. The structures of vertex algebras associated with the vacuum representations of these algebras are determined. Moreover, we prove that under certain conditions, the highest-weight irreducible modules of centrally-extended classical Lie algebras of infinite matrices with finite number of nonzero entries naturally give rise to the irreducible modules of the simple quotients of these vertex algebras. From vertex algebra and its representation point of view, our results with positive integral central charge are high-order differential operator analogues of the well-known WZW models in conformal field theory associated with affine Kac-Moody algebras. Indeed, when the left ideals are the algebra of differential operators, our Lie algebras do contain affine Kac-Moody algebras as subalgebras and our results restricted on them are exactly the representation contents in WZW models. Similar results with negative central charge are also obtained.     

Conformal designs based on vertex operator algebras

We introduce the notion of a conformal design based on a vertex operator algebra. This notation is a natural analog of the notion of block designs or spherical designs when the elements of the design are based on self-orthogonal binary codes or integral lattices, respectively. It is shown that the subspaces of an extremal self-dual vertex operator algebra of fixed degree form conformal 11-, 7-, or 3-designs, generalizing similar results of Assmus and Mattson and Venkov for extremal doubly-even codes and extremal even lattices. Other examples are coming from group actions on vertex operator algebras, the case studied first by Matsuo. The classification of conformal 6- and 8-designs is investigated. Again, our results are analogous to similar results for codes and lattices.