Hochschild homology of certain Soergel bimodules

The Soergel bimodules were introduced by Soergel in [9, 10] in the context of the infinite-dimensional representation theory of simple Lie algebra and Kazhdan-Lusztig theory. They have nice explicit expression as the tensor products of the rings of polynomials invariant under the action of a symmetric group, tensored over rings of the same form. Moreover, there are various quite different interpretations of the Soergel bimodules (e.g. the geometric interpretation [11]), that make them very important. Recently, Soergel bimodules were used by Rouquier [8] to categorify braid groups. Khovanov [3] extended this construction by adding Hochschild homology of the Soergel bimodules involved, to obtain a link invariant — the triply-graded categorification of the HOMFLY-PT polynomial. The Hochschild homology works perfectly to give invariance under the Markov moves (Reidemeister 1 move). Also, Khovanov has shown that this construction is isomorphic to the previous one by Khovanov and Rozansky [5], that uses the Koszul complexes. Moreover, in [6], we have extended the Khovanov’s construction for the categorification of 1,2colored HOMFLY-PT polynomial. A larger set of Soergel bimodules was used in the definition, and the computations proving the invariance under the Reidemeister moves are much more involved. Also, we have sketched the construction in the general case i.e. for the categorification of arbitrary colored HOMFLY-PT polynomial. However, because of the difficulty of the explicit expressions, the proof of the invariance was postponed for the subsequent paper.