2018/07/25 by Entova-Aizenbud, Inna, Serganova, Vera
#FOS: Mathematics #Representation Theory (math.RT)
paper · doi:10.48550/arxiv.1807.09478
We study stabilization of finite-dimensional representations of the periplectic Lie superalgebras \mathfrakp(n) as n → ∞. The paper gives a construction of the tensor category Rep(\underlineP), possessing nice universal properties among tensor categories over the category \mathttsVect 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 arXiv:1511.07699. Secondly, given a tensor category C over \mathttsVect, exact tensor functors Rep(\underlineP)\longrightarrow 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(\underlineP) is constructed in two ways. The first construction is through an explicit limit of the tensor categories Rep(\mathfrakp(n)) (n≥ 1) under Duflo-Serganova functors. The second construction (inspired by P. Etingof) describes Rep(\underlineP) as the category of representations of a periplectic Lie supergroup in the Deligne category \mathttsVect \boxtimes Rep(\underlineGLt). An upcoming paper by the authors will give results on the abelian and tensor structure of Rep(\underlineP).