Quantum Vertex Algebras And Their Representations

In this thesis, we study invariant bilinear forms on M(o|¨)bius nonlocal vertex alge-bras, regular representations of M(o|¨)bius quantum vertex algebras, and vertex algebrasassociated with elliptic affine Lie algebras.I. Frenkel, J. Lepowsky and Y. Huang [FHL] studied invariant bilinear forms onvertex operator algebras the first time. H. Li [Li1] has systematically studied sym-metric invariant bilinear forms on vertex operator algebras and gave a criterion fordetermining the existence of nonzero symmetric invariant bilinear forms on vertex op-erator algebras. In [Sc], N. R. Scheithauer studied invariant bilinear forms on vertexsuperalgebras with Virasoro element. Invariant bilinear forms on vertex algebras andvertex operator super algebras have been studied by M. Roitman [R] and H. Tamanoi[T] respectively. In Chapter 3, we define the invariant bilinear forms on M(o|¨)bius nonlo-cal vertex algebras slightly different and give a criterion for determining the existenceof nonzero invariant bilinear forms on M(o|¨)bius nonlocal vertex algebras similarly. Ourfirst main result is the following theorem.Theorem 1: Let V be a M(o|¨)bius nonlocal vertex algebra, and let f be the linearfunctional on V(0). And we extend f to be a linear function on V by defining f(V(n)) =0, for n≠0. We define a bilinear form on V satisfies and (1,u) = f(u), for u,v∈V . The bilinear form determined by f is invariant if andonly if L(1)V(1) ? ker f. Furthermore, the space of invariant bilinear forms on V isnaturally isomorphic to the dual space of V(0)/L(1)V(1).In [Li4], Li studied regular representations of vertex operator algebras. The reg-ular representation has been used to study Zhu's A(V )-theory, induced modules [Li5]and tensor functors for vertex operator algebras [Li6]. In Chapter 4, we study regularrepresentations for M(o|¨)bius quantum vertex algebras. Given a M(o|¨)bius quantum vertexalgebra V , a V -module W and a nonzero complex number z, we define a canonical subspace DP(z)(W) of W* and the (unique) linear map YP(z)(·,x) from V (?) V to(EndDP(z)(W))[[x,x^-1]]. Our second main result is the following theorem.Theorem 2: The pair (DP(z)(W),YP(z)) carries the structure of a (V (?) V )op-module.Elliptic affine Lie algebras, similar to affine Lie algebras, are a family of infinite-dimensional Lie algebras associated with finite-dimensional simple Lie algebras. Bothelliptic affine Lie algebras and affine Lie algebras are special examples of generalKrichever-Novikov algebras ( [KN1, KN2] ). It has been long known (see [Bo1,FLM, FZ, DL]) that affine Lie algebras g(?) can be canonically associated with vertexalgebras. In Chapter 5, we study elliptic affine Lie algebras in the context of ver-tex algebras and their modules. Let g be a (possibly infinite-dimensional) Lie algebraover C and g1 be a vector space isomorphic to g. For each polynomial p(x)∈C[x],g(?)p = (g⊕g1) (?) C[t,t(?)1]⊕Ck is a Lie algebra over C which generalizes elliptic affineLie algebra g(?)e in a certain way, gˇp = C((z)) (?) (g⊕g1) (?) C[t,t(?)1]⊕C((z))k is a Liealgebra over C((z)), and Vˇgp((?), 0) is a vertex C((z))-algebra associated with gˇp and acomplex number (?). Our third main result is the following theorem.Theorem 3: For any restricted g(?)p-module W of level (?), there exists a uniquestructure YW of a type zero Vˇgp((?), 0)-module such thatOn the other hand, let (W,YW) be a type zero Vˇgp((?), 0)-module. Then W is a restrictedg(?)p-module of level (?) withand with k acting as scalar (?).