Quantized enveloping superalgebra of type $P$

We introduce a new quantized enveloping superalgebra $mathfrak{U}_q{mathfrak{p}}_n$ attached to the Lie superalgebra ${mathfrak{p}}_n$ of type $P$. The superalgebra $mathfrak{U}_q{mathfrak{p}}_n$ is a quantization of a Lie bisuperalgebra structure on ${mathfrak{p}}_n$ and we study some of its basic properties. We also introduce the periplectic $q$-Brauer algebra and prove that it is the centralizer of the $mathfrak{U}_q {mathfrak{p}}_n$-module structure on ${mathbb C}(n|n)^{otimes l}$. We end by proposing a definition for a new periplectic $q$-Schur superalgebra.