The Exponential Map for Hopf Algebras

Abstract

We give an analogue of the classical exponential map on Lie groups for Hopf ∗-algebras with differential calculus. The major difference with the classical case is the interpretation of the value of the exponential map, classically an element of the Lie group. We give interpretations as states on the Hopf algebra, elements of a Hilbert 𝐶∗-bimodule of 1/2 densities, and elements of the dual Hopf algebra. We give examples for complex-valued functions on the groups 𝑆₃ and ℤ, Woronowicz's matrix quantum group ℂq[𝑆𝑈₂], and the Sweedler-Taft algebra.