Riemannian Geometry of Quantum Groups and Finite Groups with Nonuniversal Differentials

Algebraic Approach to Quantum Gravity Iii: Noncommmutative Riemannian Geometry

This is a self-contained introduction to quantum Riemannian geometry based on quantum groups as frame groups, and its proposed role in quantum gravity. Much of the article is about the generalisation of classical Riemannian geometry that arises naturally as the classical limit; a theory with nonsymmetric metric and a skew version of metric compatibilty. Meanwhile, in quantum gravity a key ingre...

Conceptual Issues for Noncommutative Gravity on Algebras and Finite Sets

We discuss some of the issues to be addressed in arriving at a definitive noncommutative Riemannian geometry that generalises conventional geometry both to the quantum domain and to the discrete domain. This also provides an introduction to our 1997 formulation based on quantum group frame bundles. We outline now the local formulae with general differential calculus both on the base ‘quantum ma...

Classification of Differentials on Quantum Doubles and Finite Noncommutative Geometry

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 C(G)◮<CG = D∗(G) of a finite group G. We give D∗(S3) as an example. We ...

Reflection Quotients in Riemannian Geometry. a Geometric Converse to Chevalley’s Theorem

Chevalley’s theorem and it’s converse, the Sheppard-Todd theorem, assert that finite reflection groups are distinguished by the fact that the ring of invariant polynomials is freely generated. We show that in the Euclidean case, a weaker condition suffices to characterize finite reflection groups, namely that a freely-generated polynomial subring is closed with respect to the gradient product. ...

Riemannian Geometry of Quantum Groups