Symmetric invariant cocycles on the duals of q-deformations

TL;DR

This paper proves that symmetric invariant 2-cocycles on certain quantum group completions are trivial up to central coboundaries, establishing uniqueness of related Drinfeld twists and showing the spectral triple's independence from choices.

Contribution

It demonstrates the triviality of symmetric invariant 2-cocycles for non-root-of-unity q and establishes the uniqueness of Drinfeld twists up to central coboundaries.

Findings

Symmetric invariant 2-cocycles are coboundaries of central elements.

Drinfeld twists relating U_q(g) and U(g) are unique up to central coboundaries.

The spectral triple for q-deformed G is independent of choices up to unitary equivalence.

Abstract

We prove that for q not a nontrivial root of unity any symmetric invariant 2-cocycle for a completion of Uq(g) is the coboundary of a central element. Equivalently, a Drinfeld twist relating the coproducts on completions of Uq(g) and U(g) is unique up to coboundary of a central element. As an application we show that the spectral triple we defined in an earlier paper for the q-deformation of a simply connected semisimple compact Lie group G does not depend on any choices up to unitary equivalence.