Deligne categories and the periplectic Lie superalgebra

Abstract

We study stabilization of finite-dimensional representations of the periplectic Lie superalgebras $\mathfrak{p}(n)$ as $n \to \infty$. The paper gives a construction of the tensor category $Rep(\underline{P})$, possessing nice universal properties among tensor categories over the category $\mathtt{sVect}$ of finite-dimensional complex vector superspaces. First, it is the "abelian envelope" of the Deligne category corresponding to the periplectic Lie superalgebra, in the sense of arXiv:1511.07699. Secondly, given a tensor category $\mathcal{C}$ over $\mathtt{sVect}$, exact tensor functors $Rep(\underline{P})\longrightarrow \mathcal{C}$ classify pairs $(X, \omega)$ in $\mathcal{C}$ where $\omega: X \otimes X \to \Pi \mathbf{1}$ is a non-degenerate symmetric form and $X$ not annihilated by any Schur functor. The category $Rep(\underline{P})$ is constructed in two ways. The first construction is through an explicit limit of the tensor categories $Rep(\mathfrak{p}(n))$ ($n\geq 1$) under Duflo-Serganova functors. The second construction (inspired by P. Etingof) describes $Rep(\underline{P})$ as the category of representations of a periplectic Lie supergroup in the Deligne category $\mathtt{sVect} \boxtimes Rep(\underline{GL}_t)$. An upcoming paper by the authors will give results on the abelian and tensor structure of $Rep(\underline{P})$.