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Semidirect products of C*-quantum groups: multiplicative unitaries approach

Ralf Meyer, Sutanu Roy, Stanisław Lech Woronowicz

Abstract

C*-quantum groups with projection are the noncommutative analogues of semidirect products of groups. Radford's Theorem about Hopf algebras with projection suggests that any C*quantum group with projection decomposes uniquely into an ordinary C*-quantum group and a "braided" C*-quantum group. We establish this on the level of manageable multiplicative unitaries.

Semidirect products of C*-quantum groups: multiplicative unitaries approach

Abstract

C*-quantum groups with projection are the noncommutative analogues of semidirect products of groups. Radford's Theorem about Hopf algebras with projection suggests that any C*quantum group with projection decomposes uniquely into an ordinary C*-quantum group and a "braided" C*-quantum group. We establish this on the level of manageable multiplicative unitaries.

Paper Structure

This paper contains 9 sections, 22 theorems, 137 equations.

Table of Contents

  1. Introduction
  2. Projections on Quantum Groups
  3. Semidirect products
  4. Braided Multiplicative Unitaries
  5. Semidirect product multiplicative unitaries
  6. Analysis of a quantum group with projection
  7. Examples of Quantum Groups with Projections
  8. Simplified quantum E(2) groups
  9. Some Manageability Techniques

Key Result

Proposition 2.2

Let $(C,\Delta_{C})$ be a commutative $\textup{C}^*$-quantum group with projection. Then $C\cong \textup{C}_0(G\ltimes H)$ for a semidirect product group, with the corresponding comultiplication, and the projection on $C$ comes from the group homomorphism $G\ltimes H\to G\ltimes H$, $(g,h)\mapsto (g

Theorems & Definitions (46)

  • Definition 2.1
  • Proposition 2.2
  • proof
  • Proposition 2.3
  • proof
  • Proposition 2.4
  • Proposition 2.5
  • proof
  • Lemma 2.6
  • proof
  • ...and 36 more