Cross Product Quantisation, Nonabelian Cohomology And Twisting Of Hopf Algebras

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.

26 pages

Cited by in corpus (10)

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• Translation map in quantum principal bundles
• $\kappa$-Minkowski Spacetimes and DSR Algebras: Fresh Look and Old Problems
• Braided Drinfeld and Heisenberg Doubles
• Twisted bialgebroids versus bialgebroids from a Drinfeld twist
• Twisting cocycles in fundamental representation and triangular bicrossproduct Hopf algebras
• Some remarks on the {$q$}-Poincare algebra in R-matrix form
• Frame, cotangent and tangent bundles of the quantum plane
• Finite noncommutative geometries related to $F_p[x]$
• Matrices, Bratteli Diagrams and Hopf-Galois Extensions