arXiv subjects
Publications and source records attributed to Nicolas Hemelsoet.
We show that when a torus $T$ acts on a smooth variety $X$, the twisted HKR isomorphism is equivariant. The main consequence is that the Bezrukavnikov- Lachowska isomorphism, relating the Hochschild cohomology of the principal block of the small quantum group to certain sheaf cohomology groups on the Springer resolution $\widetilde{\mathcal{N}}$ , can be upgraded to a ring isomorphism by a twist.
Let $\mathfrak{u}_\zeta^\vee$ denote the small quantum group associated with a simple Lie algebra $\mathfrak{g}^\vee$ and a root of unity $\zeta$. In [9], a geometric realization of $Z(\mathfrak{u}_\zeta^\vee)^{G^\vee}$, the $G^\vee$-invariant part of the center of $\mathfrak{u}_\zeta^\vee$, was proposed. We compute the dimension of the geometric subalgebra of the center and in the case where $G=SL_n$, we study a bigraded refinement of the result.
We apply the sheaf cohomology BGG method developed by the authors and Lachowska-Qi to the computation of Hochschild cohomology groups of various blocks of the small quantum group. All our computations of the center of the corresponding block agree with the conjectures of Lachowska-Qi. In the case of the nontrivial singular block for $\mathfrak{g} = \mathfrak{sl}_3$, we obtain the H*$(\mathfrak{u}, \mathbb{C}) = \mathbb{C}[\mathcal{N}]$-module structure of HH*$(\mathfrak{u}_λ)$.
We present a computer algorithm to explicitly compute the BGG resolution and its cohomology. We give several applications, in particular computation of various sheaf cohomology groups on flag varieties. An implementation of the algorithm is available at https://github.com/RikVoorhaar/bgg-cohomology.