Bicharacter Construction of Quantum Vertex Algebras

Abstract

This thesis is a study of the axiomatics of quantum vertex algebras based on the bicharacter construction suggested by R. Borcherds in [Bor01]. One of the goals is to use the ideas of [Bor01] to incorporate the examples of quantum vertex operators in the literature ([Jin91], [Jin95], [FR92], [FR97]), in particular the quantum vertex operators describing classes of symmetric polynomials as considered by N. Jing ([Jin94b]). The bicharacter construction is a tool which hasn't been explored in this context. We develop this construction further and based on our theory we propose the notion of HD-quantum vertex algebra. We then prove that large classes of quantum vertex operators can be given in terms of a bicharacter construction, including the Jing quantum vertex operators, and thus can be incorporated in an HD-quantum vertex algebra structure. Unlike the case of classical vertex algebras it turns out that the HD-quantum vertex algebras are not necessarily complete with respect to operator product expansions. Therefore we define an object, which we call generalized vertex algebra with Hopf symmetry, which requires completeness with respect to operator product expansions as an axiom. We then prove that for particular cases of Hopf algebras we can project this generalized vertex algebra structure to the deformed chiral algebra defined by E. Frenkel and N. Reshetikhin in [FR97]. Some of the corollaries of the bicharacter construction enable us to give formulas for the braiding map and the operator product expansions for any quantum vertex operators in the deformed chiral algebra, thereby completing the description of the main example considered in [FR97].