The marked Brauer category

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.

20 pages, submitted

References in corpus (3)

• A basis theorem for the affine oriented Brauer category and its cyclotomic quotients
• The first fundamental theorem of invariant theory for the orthosymplectic super group
• A KLR Grading of the Brauer Algebras

Cited by in corpus (11)

• The representation theory of Brauer categories I: triangular categories
• A basis theorem for the degenerate affine oriented Brauer-Clifford supercategory
• Quantized enveloping superalgebra of type $P$
• The affine VW supercategory
• The quantum isomeric supercategory
• Combinatorial skeletons of 2-cobordism and annular categories with applications to equational logic
• Diagrammatics for real supergroups
• Affine Periplectic Brauer Algebras
• The blocks of the periplectic Brauer algebra in positive characteristic
• Khovanov algebras for the periplectic Lie superalgebras
• Webs of Type P