The periplectic $q$-Brauer category

We introduce the periplectic $q$-Brauer category over an integral domain of characteristic not $2$. This is a strict monoidal supercategory and can be considered as a $q$-analogue of the periplectic Brauer category. We prove that the periplectic $q$-Brauer category admits a split triangular decomposition in the sense of Brundan-Stroppel. When the ground ring is an algebraically closed field, the category of locally finite dimensional right modules for the periplectic $q$-Brauer category is an upper finite fully stratified category in the sense of Brundan and Stroppel. We prove that periplectic $q$-Brauer algebras defined in [1] are isomorphic to endomorphism algebras in the periplectic $q$-Brauer category. Furthermore, a periplectic $q$-Brauer algebra is a standardly based algebra in the sense of Du and Rui. We construct Jucys-Murphy basis for any standard module of the periplectic $q$-Brauer algebra with respect to a family of commutative elements called Jucys-Murphy elements. Via them, we classify blocks for both periplectic $q$-Brauer category and periplectic $q$-Brauer algebras in generic case. Our result shows that both periplectic $q$-Brauer category and periplectic $q$-Brauer algebras are always not semisimple over any algebraically closed field.