Finite-dimensional Hopf C*-bimodules and C*-pseudo-multiplicative unitaries

TL;DR

This paper demonstrates the equivalence between different frameworks for describing finite quantum groupoids, connecting algebraic, bimodule, and unitary approaches in finite dimensions.

Contribution

It establishes the equivalence between Hopf-von Neumann bimodules and concrete Hopf-C*-bimodules, as well as pseudo-multiplicative unitaries and C*-pseudo-multiplicative unitaries, in finite dimensions.

Findings

Finite-dimensional Hopf-von Neumann bimodules are equivalent to concrete Hopf-C*-bimodules.

Finite-dimensional pseudo-multiplicative unitaries coincide with C*-pseudo-multiplicative unitaries.

Unified framework simplifies the understanding of finite quantum groupoids.

Abstract

Finite quantum groupoids can be described in many equivalent ways: In terms of the weak Hopf C*-algebras of B\"ohm, Nill, and Szlach\'anyi or the finite-dimensional Hopf-von Neumann bimodules of Vallin, and in terms of finite-dimensional multiplicative partial isometries or the finite-dimensional pseudo-multiplicative unitaries of Vallin. In this short note, we show that in finite dimensions, the notions of a Hopf-von Neumann bimodule and of a pseudo-multiplicative unitary coincide with the notions of a concrete Hopf-C*-bimodule and of a C*-pseudo-multiplicative unitary, respectively.