The small quantum group and the Springer resolution

Details The small quantum group and the Springer resolution The small quantum group and the Springer resolution

2006-09-28

arxiv.org/abs/math/0609819v2

Mathematics

Representation Theory

MPIM preprint, to appear in Proceedings of Haifa Conference on Quantum Groups; the second version contains minor corrections

Scientific paper

In math.RT/0304173 the derived category of the principal block in modules over the Lusztig quantum algebra at a root of unity is related to the derived category of equivariant coherent sheaves on the Springer resolution. In the present paper we deduce a similar relation between the derived category of the principal block for the {\em small (reduced) quantum algebra} and the derived category of (non-equivariant) coherent sheaves on the Springer resolution. As an application we get a geometric description of Hochschild cohomology (in particular, the center) of the regular block for the small quantum group, and use it to give an explicit description of a certain subalgebra in the center (obtained previously by another method and under more restrictive assumptions in math.QA/0107098. We also briefly explain the relation of our result to the geometric description math.RT/0407048 of the derived category of modules over the De Concini -- Kac quantum algebra.

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