JournalsprimsVol. 51, No. 1pp. 59–130

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

Abstract

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