  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.
