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

### Saeid Azam

University of Isfahan, Iran### Hiroyuki Yamane

University of Toyama, Japan### Malihe Yousofzadeh

University of Isfahan, Iran

## 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 dimU^+(\chi) = \infty, $\| \mathbb R^+(\chi)| < \infty$, 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 $≥\mathfrak g$ is B$(m, n)$ or D$(m, n)$.

## Cite this article

Saeid Azam, Hiroyuki Yamane, Malihe Yousofzadeh, Classification of Finite-Dimensional Irreducible Representations of Generalized Quantum Groups via Weyl Groupoids. Publ. Res. Inst. Math. Sci. 51 (2015), no. 1, pp. 59–130

DOI 10.4171/PRIMS/149