Cross Product Quantisation, Nonabelian Cohomology And Twisting Of Hopf Algebras

TL;DR

This paper explores the generalization of Mackey's quantization approach to quantum groups, introducing nonabelian cohomology and twisting of Hopf algebras, with implications for quantum gauge theory and topological quantum numbers.

Contribution

It develops a framework connecting nonabelian cohomology, bicrossproduct models, and twisting of Hopf algebras, extending quantum group quantization methods.

Findings

Classification of Hopf algebra extensions via nonabelian cohomology

Relation of cohomology to Drinfeld's twisting theory

Interpretation of cross product quantizations as quantum principal bundles

Abstract

This is an introduction to work on the generalisation to quantum groups of Mackey's approach to quantisation on homogeneous spaces. We recall the bicrossproduct models of the author, which generalise the quantum double. We describe the general extension theory of Hopf algebras and the nonAbelian cohomology spaces $\CH^2(H,A)$ which classify them. They form a new kind of topological quantum number in physics which is visible only in the quantum world. These same cross product quantisations can also be viewed as trivial quantum principal bundles in quantum group gauge theory. We also relate this nonAbelian cohomology $\CH^2(H,\C )$ to Drinfeld's theory of twisting.