Weak C^*-Hopf Algebras and Multiplicative Isometries

TL;DR

This paper encodes finite dimensional weak C^*-Hopf algebras into pairs consisting of a Hilbert space and a partial isometry satisfying the pentagon equation, connecting to multiplicative unitaries and quantum group frameworks.

Contribution

It introduces a novel encoding of weak C^*-Hopf algebras via pairs (H,V) and explores their relation to multiplicative unitaries and existing approaches.

Findings

Encoding of weak C^*-Hopf algebras into (H,V) pairs.

Connection between partial isometries and multiplicative unitaries.

Relation to pseudo-multiplicative unitaries by Vallin and Enock.

Abstract

We show how the data of a finite dimensional weak C^*-Hopf algebra can be encoded into a pair (H,V) where H is a finite dimensional Hilbert space and V: H \o H --> H \o H is a partial isometry satisfying, among others, the pentagon equation. In case of V being unitary we recover the Baaj-Skandalis multiplicative unitary of the discrete compact type. Relation to the pseudo- multiplicative unitary approach proposed by J.-M. Vallin and M. Enock is also discussed.