Periplectic and Isomeric Lie superalgebras, KLR algebras and Categorification

Abstract

We explore the finite dimensional representation theory of the periplectic Lie superalgebra and the isomeric Lie superalgebr, focusing on explicit descriptions of their projective generators' endomorphism rings. Using Schur–Weyl duality and KLR algebras, we establish basis theorems for cyclotomic quotients and introduce diagrammatic algebras resembling Khovanov arc algebras to obtain explicit descriptions of the endomorphism ring of a projective generator. These algebras enable explicit descriptions of translation functors and their effects on special module classes such as projective, (co)standard and irreducible modules. We conclude with categorification results: the periplectic Lie Superalgebra connects to a Fock space of the quantum electrical algebra, while the isomeric Lie superaglebra categorifies a tensor product of type B spin representations.