Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Cocycle twisting of semidirect products and transmutation

Erik Habbestad, Sergey Neshveyev

Abstract

We apply Majid's transmutation procedure to Hopf algebra maps $H \to \mathbb C[T]$, where $T$ is a compact abelian group, and explain how this construction gives rise to braided Hopf algebras over quotients of $T$ by subgroups that are cocentral in $H$. This allows us to unify and generalize a number of recent constructions of braided compact quantum groups, starting from the braided $SU_q(2)$ quantum group, and describe their bosonizations.

Cocycle twisting of semidirect products and transmutation

Abstract

We apply Majid's transmutation procedure to Hopf algebra maps , where is a compact abelian group, and explain how this construction gives rise to braided Hopf algebras over quotients of by subgroups that are cocentral in . This allows us to unify and generalize a number of recent constructions of braided compact quantum groups, starting from the braided quantum group, and describe their bosonizations.
Paper Structure (12 sections, 25 theorems, 145 equations)

This paper contains 12 sections, 25 theorems, 145 equations.

Table of Contents

  1. Generalities
  2. Categories of comodules
  3. Twisting and transmutation
  4. Transmutation over abelian groups
  5. Braided Hopf algebras over quotients of T
  6. Another view on Hb
  7. Braided compact matrix quantum groups
  8. Examples: transmuting matrix quantum groups
  9. Braided SUq(2)
  10. Braided free orthogonal quantum groups
  11. Braided free unitary quantum groups
  12. Anyonic quantum permutation groups

Key Result

Proposition 1.2

Let $A \in \mathrm{Hopf}^*(H,R)$. Then the category of $A$-comodules is isomorphic to the category of $(H \# A)$-comodules through the assignment The inverse is given by

Theorems & Definitions (53)

  • Definition 1.1
  • Proposition 1.2: cf. Majid
  • Proposition 1.3
  • Proposition 1.4: Majid-bg
  • Theorem 1.5
  • proof
  • Remark 1.6
  • Proposition 2.1
  • proof
  • Remark 2.2
  • ...and 43 more