Quadratic differential operators, Bicharacters and $\bullet$ Products

TL;DR

This paper explores the mathematical relationship between quadratic differential operators, bicharacters, and twisted products in the context of Hopf algebras, inspired by vertex operator algebra theory.

Contribution

It establishes a new connection between $ullet$ products and exponentials of quadratic differential operators using bicharacters in Hopf algebras.

Findings

Derived a relationship linking bicharacter-defined maps and quadratic differential operators.

Connected the theory to vertex operator algebra constructions.

Provided a framework for twisted module analysis in lattice vertex algebras.

Abstract

For a commutative cocommutative Hopf algebra we study the relationship between a certain linear map defined via a bicharacter, an exponential of a quadratic differential operator and a $\bullet$ product obtained via twisting by a bicharacter. This new relationship between $\bullet$ products and exponentials of quadratic differential operators was inspired by studying the exponential of a particular quadratic differential operator introduced by I. Frenkel, Lepowsky and Meurman in "Vertex operator algebras and the Monster", and used in the theory of twisted modules of lattice vertex algebras.