Koszul gradings on Brauer algebras

We show that the Brauer algebra over the complex numbers for an integral parameter delta can be equipped with a grading, in the case of delta being non-zero turning it into a graded quasi-hereditary algebra. In which case it is Morita equivalent to a Koszul algebra. This is done by realizing the Brauer algebra as an idempotent truncation of a certain level two VW-algebra for some large positive integral parameter N. The parameter delta appears then in the choice of a cyclotomic quotient. This cyclotomic VW-algebra arises naturally as an endomorphism algebra of a certain projective module in parabolic category O for an even special orthogonal Lie algebra. In particular, the graded decomposition numbers are given by the associated parabolic Kazhdan-Lusztig polynomials.Comment: 28 page

Relative cellular algebras

In this paper we generalize cellular algebras by allowing different partial orderings relative to fixed idempotents. For these relative cellular algebras we classify and construct simple modules, and we obtain other characterizations in analogy to cellular algebras. We also give several examples of algebras that are relative cellular, but not cellular. Most prominently, the restricted enveloping algebra and the small quantum group for $\mathfrak{sl}_{2}$, and an annular version of arc algebras.Comment: 39 pages, many figures, revised version, to appear in Transform. Groups, comments welcom

Schur-Weyl duality for the Brauer algebra and the ortho-symplectic Lie superalgebra

We give a proof of a Schur-Weyl duality statement between the Brauer algebra and the ortho-symplectic Lie superalgebra $\mathfrak{osp}(V)$.Comment: 22 pages, minor changes, to appear in M

Diagrams for perverse sheaves on isotropic Grassmannians and the supergroup SOSP(m|2n)

We describe diagrammatically a positively graded Koszul algebra \mathbb{D}_k such that the category of finite dimensional \mathbb{D}_k-modules is equivalent to the category of perverse sheaves on the isotropic Grassmannian of type D_k constructible with respect to the Schubert stratification. The connection is given by an explicit isomorphism to the endomorphism algebra of a projective generator described in by Braden. The algebra is obtained by a "folding" procedure from the generalized Khovanov arc algebras. We relate this algebra to the category of finite dimensional representations of the orthosymplectic supergroups. The proposed equivalence of categories gives a concrete description of the categories of finite dimensional SOSP(m|2n)-modules

Functoriality of colored link homologies

We prove that the bigraded colored Khovanov-Rozansky type A link and tangle invariants are functorial with respect to link and tangle cobordisms.Comment: 41 pages, many colored figures, some changes following suggestions of a referee, to appear in Proc. Lond. Math. Soc., comments welcom