Koszul gradings on Brauer algebras

TL;DR

This paper demonstrates that Brauer algebras over complex numbers with non-zero parameters can be endowed with a grading, making them Morita equivalent to Koszul algebras through connections with VW-algebras and category O.

Contribution

It introduces a grading on Brauer algebras via their realization as truncations of VW-algebras, establishing their Koszulity and linking to category O representations.

Findings

Brauer algebra can be graded as a quasi-hereditary algebra.

The graded decomposition numbers correspond to parabolic Kazhdan-Lusztig polynomials.

Brauer algebra is Morita equivalent to a Koszul algebra.

Abstract

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.