A polar Brauer category and Lie superalgebra representations

Abstract

We introduce a diagram category, study its structure, and investigate some of its applications to the representation theory of Lie algebras and Lie superalgebras. The morphisms of the category, which contains a subcategory isomorphic to the Brauer category, are linear combinations of `polar enhancements' of Brauer diagrams. The endomorphism algebra of each of its objects is a quotient of an algebra of chord diagrams. Analogues of the affine Temperley-Lieb category and Temperley-Lieb category of type B, whose structures are thoroughly understood, arise from particular quotients of our category. We construct a functor from our category to the full subcategory of modules for the Lie superalgebra $\mathfrak{osp}(V; \omega)$ with objects $M\otimes V^{\otimes r}$ for all $r=0, 1, \dots$, where $M$ is an arbitrary module, and $V$ is the natural module. When $M$ is the universal enveloping superalgebra $\text{U}(\mathfrak{osp}(V; \omega))$, this functor provides an effective tool for the study of $\text{U}(\mathfrak{osp}(V; \omega))$. An analysis of this functor leads to a diagrammatic construction of explicit generators for the centre of the universal enveloping superalgebra and, in the special cases when $V$ is purely even or purely odd (i.e. the classical cases), categorical interpretations of certain widely studied ``characteristic identities'' of the orthogonal and symplectic Lie algebras. In the case $V=\mathbb{C}^{0|2}$ so that $\mathfrak{osp}(V; \omega))=\mathfrak{sp}_2(\mathbb{C})$, we prove that our type B Temperley-Lieb category is isomorphic to a full subcategory of category $\mathcal O$ for $\mathfrak{sp}_2(\mathbb{C})$.