Deligne Categories and the Periplectic Lie Superalgebra

We study stabilization of finite-dimensional representations the periplectic Lie superalgebras $\mathfrak{p}(n)$ as $n \to \infty$. The paper gives a construction tensor category $Rep(\underline{P})$, possessing nice universal properties among categories over $\mathtt{sVect}$ complex vector superspaces. First, it is abelian envelope Deligne corresponding to superalgebra, in sense arXiv:1511.07699. Secondly, given $\mathcal{C}$ $\mathtt{sVect}$, exact functors $Rep(\underline{P})\longrightarrow \mathcal{C}$ classify pairs $(X, \omega)$ where $\omega: X \otimes \Pi \mathbf{1}$ non-degenerate symmetric form and $X$ not annihilated by any Schur functor. $Rep(\underline{P})$ constructed two ways. The first through an explicit limit $Rep(\mathfrak{p}(n))$ ($n\geq 1$) under Duflo-Serganova functors. second (inspired P. Etingof) describes supergroup $\mathtt{sVect} \boxtimes Rep(\underline{GL}_t)$. An upcoming authors will give results on structure $Rep(\underline{P})$.