Webs of Type P

This paper introduces type P web supercategories. They are defined as diagrammatic monoidal $k$-linear supercategories via generators and relations. We study the structure of these categories and provide diagrammatic bases for their morphism spaces. We also prove these supercategories provide combinatorial models for the monoidal supercategory generated by the symmetric powers of the natural module and their duals for the Lie superalgebra of type P.Comment: Final version. Compared to the first version, there are no substantive changes to the mathematics, but numerous changes to exposition in response to referee suggestions. Updated authors' affiliation

A basis theorem for the degenerate affine oriented Brauer-Clifford supercategory

We introduce the oriented Brauer-Clifford and degenerate affine oriented Brauer-Clifford supercategories. These are diagrammatically defined monoidal supercategories which provide combinatorial models for certain natural monoidal supercategories of supermodules and endosuperfunctors, respectively, for the Lie superalgebras of type Q. Our main results are basis theorems for these diagram supercategories. We also discuss connections and applications to the representation theory of the Lie superalgebra of type Q.Comment: 37 pages, many figures. Version 3 replaces the partial results from the previous versions with a proof by the first author of a basis theorem for cyclotomic quotients at all levels. Various other minor corrections and revisions were mad

On calibrated representations of the degenerate affine periplectic Brauer algebra

We initiate the representation theory of the degenerate affine periplectic Brauer algebra on $n$ strands by constructing its finite-dimensional calibrated representations when $n=2$. We show that any such representation that is indecomposable and does not factor through a representation of the degenerate affine Hecke algebra occurs as an extension of two semisimple representations with one-dimensional composition factors; and furthermore, we classify such representations with regular eigenvalues up to isomorphism

Quantized enveloping superalgebra of type PP

We introduce a new quantized enveloping superalgebra $\mathfrak{U}_q{\mathfrak{p}}_n$ attached to the Lie superalgebra ${\mathfrak{p}}_n$ of type $P$. The superalgebra $\mathfrak{U}_q{\mathfrak{p}}_n$ is a quantization of a Lie bisuperalgebra structure on ${\mathfrak{p}}_n$ and we study some of its basic properties. We also introduce the periplectic $q$-Brauer algebra and prove that it is the centralizer of the $\mathfrak{U}_q {\mathfrak{p}}_n$-module structure on ${\mathbb C}(n|n)^{\otimes l}$. We end by proposing a definition for a new periplectic $q$-Schur superalgebra.Comment: 14 page

The periplectic Brauer algebra

We study the periplectic Brauer algebra introduced by Moon in the study of invariant theory for periplectic Lie superalgebras. We determine when the algebra is quasi-hereditary, when it admits a quasi-hereditary 1-cover and, for fields of characteristic zero, describes the block decomposition. To achieve this, we also develop theories of Jucys-Murphy elements, Bratteli diagrams, Murphy bases, obtain a Humphreys-BGG reciprocity relation and determine some decomposition multiplicities of cell modules. As an application, we determine the blocks in the category of finite dimensional integrable modules of the periplectic Lie superalgebra