Classification of Finite-Dimensional Irreducible Representations of Generalized Quantum Groups via Weyl Groupoids

Abstract

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