Representations of the Lie superalgebra $P(n)$ and Brauer algebras with signs

The “strange” Lie superalgebra $P(n)$ is the algebra of endomorphisms of an $(n|n)$-dimensional vector space $V$ equipped with a non-degenerate odd symmetric form. The centralizer of the $P(n)$-action in the $k$-th tensor power of $V$ is given by a certain analogue of the Brauer algebra.
We discuss some properties of this algebra in application to representation theory of $P(n)$ and $P(\infty)$.
We also construct a universal tensor category such that for all n the categories of $P(n)$ modules can be obtained as quotients of this category. In some sense this category is an analogue of the Deligne categories $GL(t)$ and $SO(t)$.