Highest Weight Modules Over The Quantum Periplectic Superalgebra of Type $P$

In this paper, we begin the study of highest weight representations of the quantized enveloping superalgebra ${\mathfrak{U}}_q{\mathfrak{p}}_n$ of type $P$. We introduce a Drinfeld-Jimbo representation and establish a triangular-decomposition of ${\mathfrak{U}}_q{\mathfrak{p}}_n$. We explain how to relate modules over ${\mathfrak{U}}_q{\mathfrak{p}}_n$ to modules over ${\mathfrak{p}}_n$, the Lie superalgebra of type $P$, and we prove that the category of tensor modules over ${\mathfrak{U}}_q{\mathfrak{p}}_n$ is not semisimple.