Deligne categories and the periplectic Lie superalgebra

Abstract

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 ), possessing 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, in the sense of [EHS15].Secondly, given a tensor category C over sVect, exact tensor functors Rep(P ) → C classify pairs (X, ω) in C where ω : X ⊗ X → Π½ 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 (inspired by P. Etingof) describes Rep(P ) as the category of representations of a periplectic Lie supergroup in the Deligne category sVect ⊠ Rep(GL t ).An upcoming paper will give results on the abelian and tensor structure of Rep(P ).Contents. Introduction 1 2. Preliminaries and notation 4 3. The infinite periplectic Lie superalgebra 9 4. The subcategories Rep k (p(n)) 12 5. The DS functor 17 6. Construction of the category Rep(P ) 22 7. The first universal property 26 8. The second universal property 29 9. Lower highest weight structure 36 References 39 10. Appendix A: Direct summands of tensor powers 40 11. Appendix B: Affine group schemes in tensor categories 43

Ringel duals of Brauer algebras via super groups

“…Proof. In [EnS,Propositions 5.2.1 and 9.2.3] it is proved that Rep (r) Pe(V ) is a highest weight category where the tilting modules are precisely the direct summands of…”

The Duflo-Serganova functor, vingt ans après

“…It was used in [S2] to prove the Kac-Wakimoto conjecture (see Section 7); in [IRS] to describe the supercharacter ring for p(n) (see Section 8); in [HPS] to study important sl(∞)-modules (see Section 9); in [ES2] to give a formula for the superdimension of p(n)-modules; in [HsW] to give a new proof of the superdimension formula for GL(m|n)-modules; in [Hs] to obtain reductive envelopes of certain supergroups; in [BKN2] to compute complexity of certain modules over gl(m|n); in [CH] to classify the indecomposable summands of tensor powers of the standard representation of OSP (m|2n); and in [EHS] to construct universal tensor categories. The DS functor has been applied to study Deligne categories in numerous papers (see e.g., [CH,EhSt2,EHS,ES1]).…”

Quantized enveloping superalgebra of type $P$

“…The study of the representations of type P Lie superalgebras, which are also called periplectic in the literature, has attracted considerable attention in the last five years. Interesting results on the category O, the associated periplectic Brauer algebras, and related theories have been established in [BDEA + 1], [BDEA + 2], [CP], [Co], [CE1], [CE2], [DHIN], [EAS1], [EAS2], [KT], [Ser], among others.…”