The quantum $\epsilon$lectrical Hopf algebra and categorification of Fock space

Abstract

In this article we introduce a generalization of the Khovanov--Lauda Rouquier algebras, the electric KLR algebras. These are superalgebras which connect to super Brauer algebras in the same way as ordinary KLR-algebras of type $A$ connect to symmetric group algebras. As super Brauer algebras are in Schur--Weyl duality with the periplectic Lie superalgebras, the new algebras describe morphisms between refined translation functors for this least understood family of classical Lie superalgebras with reductive even part. The electric KLR algebras are different from quiver Hecke superalgebras introduced by Kang--Kashiwara--Tsuchioka and do not categorify quantum groups. We show that they categorify a quantum version of a type $A$ electric Lie algebra. The electrical Lie algebras arose from the study of electrical networks. They recently appeared in the mathematical literature as Lie algebras of a new kind of electric Lie groups introduced by Lam and Pylyavskyy. We give a definition of a quantum electric algebra and realise it as a coideal subalgebra in some quantum group. We finally prove several categorification theorems: most prominently we use cyclotomic quotients of electric KLR algebras to categorify higher level Fock spaces.