Narrow Results

We consider a class of bimodules over polynomial algebras which were originally introduced by Soergel in relation to the Kazhdan–Lusztig theory, and which describe a direct summand of the category of Harish–Chandra modules for sl(n). Rouquier used Soergel bimodules to construct a braid group action on the homotopy category of complexes of modules over a polynomial algebra. We apply Hochschild homology to Rouquier's complexes and produce triply-graded homology groups associated to a braid. These groups turn out to be isomorphic to the groups previously defined by Lev Rozansky and the author, which depend, up to isomorphism and overall shift, only on the closure of the braid. Consequently, our construction produces a homology theory for links.

We define virtual braid groups of type B and construct a morphism from such a group to the group of isomorphism classes of some invertible complexes of bimodules up to homotopy.

Diagrammatic algebra provides a useful way to study Soergel bimodules. This approach proceeds via the simpler category $𝔹𝕊$Bim of Bott–Samelson bimodules, for which there is a well developed diagrammatic calculus. As Soergel bimodules are summands of Bott–Samelson bimodules, it is important to understand idempotents in the category $𝔹𝕊$Bim.

For Coxeter groups of type $B$, we analyze this problem for certain important idempotents, namely, the idempotent projecting to the indecomposable Soergel bimodule corresponding to the longest element of the Coxeter group. We use a strategy analogous to one used by Elias in type $A$. We present a full explicit calculation for the first nontrivial case $B_3$. The strategy is applicable for general $B$ but much more involved. We hope that the $B_3$ case serves as a stepping stone for the general case.