Kirillov–Reshetikhin Modules of Generalized Quantum Groups of Type $A$

Abstract

The generalized quantum group of type $A$ is an affine analogue of the quantum group associated to a general linear Lie superalgebra, which appears in the study of solutions to the tetrahedron equation or the three-dimensional Yang–Baxter equation. In this paper we develop the crystal base theory for finite-dimensional representations of generalized quantum groups of type $A$. As the main result, we construct Kirillov–Reshetikhin modules, that is, a family of irreducible modules which have crystal bases. We also give an explicit combinatorial description of the crystal structure of Kirillov–Reshetikhin modules, the combinatorial $R$ matrix and the energy function on their tensor products.