Super-bicharacter construction of HD-quantum vertex algebras

Abstract

We extend the bicharacter construction of quantum vertex algebras first proposed by Borcherds to the case of super Hopf algebras. We give a bicharacter description of the charged free fermion super vertex algebra, which allows us to construct different quantizations of it in the sense of H D -quantum vertex algebras, or specializations to Etingof-Kazhdan quantum vertex algebras. We give formulas for the analytic continuation of product of fields, the operator product expansion and the normal ordered product in terms of the super-bicharacters.

$H_D$-Quantum Vertex Algebras

“…An effective method to do calculations in V L,t and similar H D -quantum vertex algebras is given by the theory of bicharacters, see [Ang06].…”

HD-QUANTUM VERTEX ALGEBRAS AND BICHARACTERS

“…Also, not unrelated, the braiding map S z1,z2 in [EK00] is in fact assumed to be of the form S z1,z2 = Sz1−z2 , where Sz is a function of a single variable. It is also shown in [Ang06] that this does not holds for symmetric polynomials.…”

“…We do take, however, from Borcherds' paper the idea of a bicharacter as a method to construct examples: we will use bicharacters both to produce the vertex operators and the braiding. (See [Ang06] for more details. )…”

“…The Etingof-Kazhdan theory is very close to the classical theory, in fact so close that it is not suitable to describe quantum vertex operators related to symmetric polynomials. Briefly, in the usual theory (and in [EK00]) vertex operators a(z) satisfy translation covariance of the form It was shown in the thesis [Ang06] that (1.1) can not hold in the case of quantum vertex operators related to symmetric polynomials. Also, not unrelated, the braiding map S z1,z2 in [EK00] is in fact assumed to be of the form S z1,z2 = Sz1−z2 , where Sz is a function of a single variable.…”

“…It was shown in the thesis [Ang06] that (1.1) can not hold in the case of quantum vertex operators related to symmetric polynomials. Also, not unrelated, the braiding map S z1,z2 in [EK00] is in fact assumed to be of the form S z1,z2 =S z1−z2 , whereS z is a function of a single variable.…”