Quadratic differential operators, Bicharacters and $\bullet$ Products

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.