Classification of differentials on quantum doubles and finite noncommutative geometry

Details Classification of differentials on quantum doubles and finite noncommutative geometry Classification of differentials on quantum doubles and finite noncommutative geometry

2002-05-14

arxiv.org/abs/math/0205150v2

Mathematics

Quantum Algebra

Final version to be published Marcel Dekker Lect. Notes Pure Appl. Maths; added final section with calculus on quantum double

Scientific paper

We discuss the construction of finite noncommutative geometries on Hopf algebras and finite groups in the `quantum groups approach'. We apply the author's previous classification theorem, implying that calculi in the factorisable case correspond to blocks in the dual, to classify differential calculi on the quantum codouble $D^*(G)=kG\lcocross k(G)$ of a finite group $G$. We give $D^*(S_3)$ as an example including its exterior algebra and lower cohomology. We also study the calculus on $D^*(\CA)$ induced from one on a general Hopf algebra $\CA$ in general and specialise to $D^*(G)=U(\cg)\lcocross k[G]$ as a noncommutative isometry group of an enveloping algebra $U(\cg)$ as a noncommutative space.

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