Quantum differentials on U_q(g) and q-fuzzy spheres

Shahn Majid (Queen Mary)

Wednesday 4th November, 2009 16:00-17:00 Mathematics Building, Room 204


Just as the enveloping algebra U(g) of a Lie algebra g can be viewed as a quantisation of g*, we consider quantum groups U_q(g) as a quantisation or `coordinate algebra'. We pose the question of their natural noncommutative differential calculus in the sense of a differential graded algebra, and find a canonical answer using braided category methods. We obtain explicit results for the case of the algebra U_q(su_2) and the q-hyperboloid of which it is a localisation, and show that the noncommutative differential geometry of the latter is obtained systematically from the noncommutative differential geometry of C_q(SU_2) by a (twist)-equivalence of comodule categories. The non-standard Podles spheres naturally turn out to be cross-sections of this same q-hyperboloid and we provide a braided category approach their construction. If time, recent progress on some other noncommutative differential geometries of mathematical interest will be covered.

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