The marked Brauer category

TL;DR

This paper introduces the marked Brauer algebra and category, extending classical concepts to superalgebra settings, and establishes their duality with Lie superalgebras, classifying modules and tensor decompositions.

Contribution

It generalizes the Brauer algebra to the superalgebra context, classifies simple modules, and demonstrates Schur-Weyl duality with Lie superalgebras, unifying previous results.

Findings

Classified simple modules of the marked Brauer algebra.

Established Schur-Weyl duality with Lie superalgebras.

Described indecomposable summands of tensor powers.

Abstract

We introduce the marked Brauer algebra and the marked Brauer category. These generalize the analogous constructions for the ordinary Brauer algebra to the setting of a homogeneous bilinear form on a $\mathbb{Z}_2$-graded vector space. We classify the simple modules of the marked Brauer algebra over any field of characteristic not two. Under suitable assumptions we show that the marked Brauer algebra is in Schur-Weyl duality with the Lie superalgebra, $\mathfrak{g}$, of linear maps which leave the bilinear form invariant. We also provide a classification of the indecomposable summands of the tensor powers of the natural representation for $\mathfrak{g}$ under those same assumptions. In particular, our results generalize Moon's work on the Lie superalgebra of type $\mathfrak{p}(n)$ and provide a unifying conceptual explanation for his results.