Moscow Mathematical Journal

Volume 21, Issue 3, July–September 2021 pp. 507–565.

Deligne Categories and the Periplectic Lie Superalgebra

Authors: Inna Entova-Aizenbud (1) and Vera Serganova (2)
Author institution:(1) Dept. of Mathematics, Ben Gurion University, Beer-Sheva, Israel
(2) Dept. of Mathematics, University of California at Berkeley, Berkeley, CA 94720

Summary:

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 $\mathrm{Rep}(\underline{P})$ , 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.

Secondly, given a tensor category $\mathcal{C}$ over $\mathtt{sVect}$ , exact tensor functors $\mathrm{Rep}(\underline{P})\rightarrow \mathcal{C}$ classify pairs $(X,\omega)$ in $\mathcal{C}$ , where $\omega\colon X \otimes X \to \Pi 1$ is a non-degenerate symmetric form and $X$ not annihilated by any Schur functor.

The category $\mathrm{Rep}(\underline{P})$ is constructed in two ways. The first construction is through an explicit limit of the tensor categories $\mathrm{Rep}(\mathfrak{p}(n))$ ( $n\geq 1$ ) under Duflo–Serganova functors. The second construction (inspired by P. Etingof) describes $\mathrm{Rep}(\underline{P})$ as the category of representations of a periplectic Lie supergroup in the Deligne category $\mathtt{sVect} \boxtimes \mathrm{Rep}(\underline{\mathrm{GL}}{}_t)$ .

2020 Math. Subj. Class. 17A70, 17B10, 17B20, 18D10.

Keywords: Deligne categories, periplectic Lie superalgebra, tensor categories, stabilization in representation theory, Duflo–Serganova functor.

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