The small quantum group and the Springer resolution

Abstract

In [ABG] 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 e N . In the present paper we deduce a similar relation between the derived category of the principal block for the small (reduced) quantum algebra u and the derived category of (non-equivariant) coherent sheaves on e N . As an application we get a geometric description of Hochschild cohomology (in particular, the center) of the regular block for u, and use it to give an explicit description of a certain subalgebra in the center (such a subalgebra was obtained previously by another method and under more restrictive assumptions in [La]). We also briefly explain the relation of our result to the geometric description [BK] of the derived category of modules over the De Concini -Kac quantum algebra.To the memory of Iosef Donin. Contents1. Quantum algebras 1.1. Basic notations and conventions 1.2. Quantum algebras at a root of unity 1.3. Quantum Lusztig-Frobenius map 2. Functor to a derived category of u-modules 2.1. The principal block 2.2. Springer resolution and U-modules 2.3. Springer resolution and u-modules 2.4. Connection to the Kac-De Concini algebra and a result of [BK] 3. Hochschild cohomology of the principal block of u 3.1. The result 3.2. The proof 3.3. An explicit subalgebra in the principal block of the center of the small quantum group.

The center of small quantum groups II: singular blocks

“…, and take the usual tensor product of ι * (O gP ) with the resolution over O g×X . Similarly as done in [8,Section 2.4], one computes that the sheaf of DG algebras is canonically quasiisomorphic to the sheaf of DG algebras…”

“…Below we derive a geometric description of the singular blocks of the center, analogous to the result in [8,Section 2.4]. corresponding to all positive roots α ∈ Φ + and n 1 (see [22,23]).…”

“…Geometric description of the singular blocks. In this subsection, we closely follow the arguments of [8,Section 2.4].…”

“…The paper is organized as follows. We start Section 2 with a generalization to the singular block case of two crucial results stated in [4,8]. We establish the equivalence between the singular block of the (big and the small) quantum groups, on one side, and certain categories of coherent sheaves on the parabolic Springer resolution (Theorem 2.1 and Corollary 2.2).…”

Factorizable ribbon quantum groups in logarithmic conformal field theories

“…15 The small quantum groups have been the subject of some constant interest, see, e.g., [46,47,48] and the references therein.…”