TY - JOUR

T1 - Deligne categories and the periplectic lie superalgebra

AU - Entova-Aizenbud, Inna

AU - Serganova, Vera

N1 - Funding Information:
Received April 13, 2019; in revised form November 10, 2020. The reseach of both authors in MSRI was supported by the National Science Foundation under Grant No. DMS-1440140. I.E.-A. was supported by the Israel Science Foundation (ISF grant no. 711/18). V.S. was supported by NSF grant 1701532.
Publisher Copyright:
© 2021 Independent University of Moscow.

PY - 2021/7/1

Y1 - 2021/7/1

N2 - We study stabilization of finite-dimensional representations of the periplectic Lie superalgebras p(n) as n → ∞. The paper gives a construction of the tensor category Rep(P ), pos-sessing nice universal properties among tensor categories over the category 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 C over sVect, exact tensor functors Rep(P ) → C classify pairs (X, ω) in C, where ω: X ⊗ X → Π1 is a non-degenerate symmetric form and X not annihilated by any Schur functor. The category Rep(P ) is constructed in two ways. The first construction is through an explicit limit of the tensor categories Rep(p(n)) (n ≥ 1) under Duflo–Serganova functors. The second construction (in-spired by P. Etingof ) describes Rep(P ) as the category of representations of a periplectic Lie supergroup in the Deligne category sVect⊠Rep(GLt ).

AB - We study stabilization of finite-dimensional representations of the periplectic Lie superalgebras p(n) as n → ∞. The paper gives a construction of the tensor category Rep(P ), pos-sessing nice universal properties among tensor categories over the category 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 C over sVect, exact tensor functors Rep(P ) → C classify pairs (X, ω) in C, where ω: X ⊗ X → Π1 is a non-degenerate symmetric form and X not annihilated by any Schur functor. The category Rep(P ) is constructed in two ways. The first construction is through an explicit limit of the tensor categories Rep(p(n)) (n ≥ 1) under Duflo–Serganova functors. The second construction (in-spired by P. Etingof ) describes Rep(P ) as the category of representations of a periplectic Lie supergroup in the Deligne category sVect⊠Rep(GLt ).

KW - Deligne categories

KW - Duflo–Serganova func-tor

KW - Periplectic Lie superalgebra

KW - Stabilization in representation theory

KW - Tensor categories

UR - http://www.scopus.com/inward/record.url?scp=85109897097&partnerID=8YFLogxK

U2 - 10.17323/1609-4514-2021-21-3-507-565

DO - 10.17323/1609-4514-2021-21-3-507-565

M3 - Article

AN - SCOPUS:85109897097

VL - 21

SP - 507

EP - 565

JO - Moscow Mathematical Journal

JF - Moscow Mathematical Journal

SN - 1609-3321

IS - 3

ER -