Classification of finite-dimensional irreducible representations of generalized quantum groups via weyl groupoids

Abstract

Let χ be a bi-homomorphism over an algebraically closed field of characteristic zero. Let U(χ) be a generalized quantum group, associated with χ, such that dim U+(χ) = ∞,|R+(χ) | < ∞, and R+(χ) is irreducible, where U+(χ) is the positive part of U(χ), and R+(χ) is the Kharchenko positive root system of U+(χ). In this paper, we give a list of finite-dimensional irreducible U(χ)-modules, relying on a special reduced expression of the longest element of the Weyl groupoid of R(χ) := R+(χ) ∪ (–R+(χ)). From the list, we explicitly obtain lists of finite-dimensional irreducible modules for simple Lie superalgebras g of types A-G and the (standard) quantum superalgebras Uq(g). An intrinsic gap appears between the lists for g and Uq(g), e.g, if g is B(m, n) or D(m, n).