Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 18 (2022), 017, 17 pages      arXiv:2203.04549      https://doi.org/10.3842/SIGMA.2022.017

The Exponential Map for Hopf Algebras

Ghaliah Alhamzi a and Edwin Beggs b
a) Imam Mohammad Ibn Saud Islamic University (IMSIU), Riyadh, Saudi Arabia
b) Department of Mathematics, Swansea University, Wales, UK

Received June 15, 2021, in final form February 16, 2022; Published online March 09, 2022

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.

Key words: Hopf algebra; differential calculus; Lie algebra; exponential map.

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