The Exponential Map for Hopf Algebras

TL;DR

This paper develops an analogue of the exponential map for Hopf *-algebras with differential calculus, interpreting its values as states, bimodule elements, or dual algebra elements, with examples from various quantum groups.

Contribution

It introduces a new exponential map for Hopf *-algebras and provides multiple interpretations and concrete examples, extending classical Lie group concepts to quantum algebra settings.

Findings

Defined an exponential map for Hopf *-algebras with differential calculus

Provided interpretations as states, bimodule elements, and dual algebra elements

Presented examples involving groups, quantum groups, and 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 $\frac{1}{2}$ densities and elements of the dual Hopf algebra. We give examples for complex valued functions on the groups $S_{3}$ and $\mathbb{Z}$, Woronowicz's matrix quantum group $\mathbb{C}_{q}[SU_2] $ and the Sweedler-Taft algebra.