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This article is cited in 6 scientific papers (total in 6 papers)
Deligne categories and the periplectic Lie superalgebra
Inna Entova-Aizenbuda, Vera Serganovab a Dept. of Mathematics, Ben Gurion University, Beer-Sheva, Israel
b Dept. of Mathematics, University of California at Berkeley, Berkeley, CA 94720
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 $\mathrm{Rep}(\underline{P})$, possessing nice universal properties among tensor categories over the category $\mathrm{sVect}$ 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 $\mathrm{sVect}$, exact tensor functors $\mathrm{Rep}(\underline{P})\rightarrow \mathcal{C}$ classify pairs $(X, \omega)$ in $\mathcal{C}$, where $\omega\colon X \otimes X \to \Pi1$ 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 $\mathrm{sVect} \boxtimes \mathrm{Rep}(\underline{\mathrm{GL}}_t)$.
Key words and phrases:
deligne categories, periplectic Lie superalgebra, tensor categories, stabilization in representation theory, Duflo–Serganova functor.
Citation:
Inna Entova-Aizenbud, Vera Serganova, “Deligne categories and the periplectic Lie superalgebra”, Mosc. Math. J., 21:3 (2021), 507–565
Linking options:
https://www.mathnet.ru/eng/mmj804 https://www.mathnet.ru/eng/mmj/v21/i3/p507
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Abstract page: | 60 | References: | 23 |
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