Deligne categories and the periplectic lie superalgebra

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(GL_t ).