The marked Brauer category

Abstract

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