Brian Parshall (University of Virginia)

Friday 21st July, 12.05-12.55pm, Carslaw 373

Quantum group cohomology via the geometry of the nullcone

The explicit calculation of the cohomology algebra of a restricted Lie algebra (in positive characteristic) is unknown in most cases, although the associated algebraic variety is homeomorphic to the nullcone of the Lie algebra (by Bendel-Friedlander-Suslin). When the Lie algebra is that of a semisimple group and the characteristic of the field is larger than the Coxeter number, the cohomology is explicitly known (by Friedlander-Parshall, Andersen-Jantzen). However, little information is available for small primes. We consider the similar situation for quantum enveloping algebras at a root of unity. Intuitively, the quantum case should be an easier approximation to the restricted enveloping algebra case, and we can provide fairly complete answers. The application of powerful tools from complex geometry provide at least one advantage in this case. (This talk reports on joint work with C. Bendel, D. Nakano and C. Pillen).