The periplectic \(q\)-Brauer category (Q6632084)

The periplectic Brauer algebra was introduced by Moon when he studied tensor products of the natural module for the periplectic Lie superalgebra \(\mathfrak{p}_n\). Later, Kujawa and Tharp introduced the periplectic Brauer category and proved that any periplectic Brauer algebra appears as an endomorphism algebra of a corresponding object in the periplectic Brauer category. Since then the representation theory of periplectic Brauer algebras has been studied extensively, including the classification of blocks, decomposition numbers, and the related (weak) categorification, all of which have important applications in the representation theory of \(\mathfrak{p}_n\).\N\NH. Rui and L. Song introduce the periplectic \(q\)-Brauer category over an integral domain of characteristic not equal to 2. This is a strict monoidal supercategory and can be considered as a \(q\)-analogue of the periplectic Brauer category. The authors prove that the periplectic \(q\)-Brauer category admits a split triangular decomposition. 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. They prove that periplectic \(q\)-Brauer algebras are isomorphic to endomorphism algebras in the periplectic \(q\)-Brauer category. Furthermore, a periplectic \(q\)-Brauer algebra is a standardly based algebra. Rui and Song construct a 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, which then they classify blocks for both the periplectic \(q\)-Brauer category and periplectic \(q\)-Brauer algebras in the generic case. This shows that both the periplectic \(q\)-Brauer category and periplectic \(q\)-Brauer algebras are always not semisimple over any algebraically closed field.