HD-QUANTUM VERTEX ALGEBRAS AND BICHARACTERS

Abstract

We define a new class of quantum vertex algebras, based on the Hopf algebra HD = ℂ[D] of "infinitesimal translations" generated by D. Besides the braiding map describing the obstruction to commutativity of products of vertex operators, HD-quantum vertex algebras have as a main new ingredient a "translation map" that describes the obstruction of vertex operators to satisfying translation covariance. The translation map also appears as obstruction to the state-field correspondence being a homomorphism. We use a bicharacter construction of Borcherds to construct a large class of HD-quantum vertex algebras. One particular example of this construction yields a quantum vertex algebra that contains the quantum vertex operators introduced by Jing in the theory of Hall–Littlewood polynomials.