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A Coassociative C*-Quantum Group with Non-Integral Dimensions

G. Bohm, K. Szlachanyi

Abstract

By weakening the counit and antipode axioms of a C*-Hopf algebra and allowing for the coassociative coproduct to be non-unital we obtain a quantum group, that we call a weak C*-Hopf algebra, which is sufficiently general to describe the symmetries of essentially arbitrary fusion rules. This amounts to generalizing the Baaj-Skandalis multiplicative unitaries to multipicative partial isometries. Every weak C*-Hopf algebra has a dual which is again a weak C*-Hopf algebra. An explicit example is presented with Lee-Yang fusion rules. We shortly discuss applications to amalgamated crossed products, doubles, and quantum chains.

A Coassociative C*-Quantum Group with Non-Integral Dimensions

Abstract

By weakening the counit and antipode axioms of a C*-Hopf algebra and allowing for the coassociative coproduct to be non-unital we obtain a quantum group, that we call a weak C*-Hopf algebra, which is sufficiently general to describe the symmetries of essentially arbitrary fusion rules. This amounts to generalizing the Baaj-Skandalis multiplicative unitaries to multipicative partial isometries. Every weak C*-Hopf algebra has a dual which is again a weak C*-Hopf algebra. An explicit example is presented with Lee-Yang fusion rules. We shortly discuss applications to amalgamated crossed products, doubles, and quantum chains.

Paper Structure

This paper contains 5 sections, 4 theorems, 56 equations.

Table of Contents

  1. Introduction
  2. Axioms of Weak $C^*$-Hopf Algebras
  3. The Weyl algebras and the Double
  4. Weak $C^*$-Hopf Algebras and the Pentagon Equation
  5. An Example with Lee--Yang Fusion Rules

Key Result

Theorem 2.2

If $(A,{1\!1},\Delta,\varepsilon,S)$ satisfies Axioms (A1--4) then $(\hat{A},\hat{{1\!1}},\hat{\Delta},\hat{\varepsilon}, \hat{S})$ satisfies Axioms (A1--4), too. That is the notion of a weak $^*$-Hopf algebra is selfdual.

Theorems & Definitions (5)

  • Definition 2.1
  • Theorem 2.2
  • Theorem 2.3
  • Proposition 2.4
  • Corollary 2.5