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Let U(χ) be a generalized quantum group such that dimU+(χ) = ∞, |R+(χ)| < ∞, and R+(χ) is irreducible, where U+(χ) is the positive part of U(χ), and R+(χ) is the Kharchenko’s positive root system of U+(χ). In this paper, we give a list of finite-dimensional irreducible highest weight U(χ)-modules, relying on a special reduced expression of the longest element of the Weyl groupoid of U(χ).

In this paper, we give a classification of the finite dimensional irreducible representations of a nonstandard quantum group U, which may be viewed as a Z/3Z-graded version of a quantum superalgebra.

1 Preliminary. Keep the notation and terminology of [1]. We say that the datum (E,Π, p) is of (A(1, 1)(1))H type if (E,Π, p) is of affine ABCD type (see Definition 1.4.1), Π = {α0, α1, α2, α3} and if the Dynkin diagram of (E,Π, p) is either of the two Dynkin diagrams of (1). Until the end of this section, we assume that (E,Π, p) is of (A(1, 1)(1))H type. Then G(E,Π, p) ∼= (A(1, 1)(1))H, G (E,Π, p) ∼= (sl(2, 2)(1))H, and Φ(E,Π, p) = {±(m+1)δ, ±(mδ+αi), ±(mδ+ αi + αi−1), ±(mδ + αi + αi−1 + αi−2)|i = 0, 1, 2, 3 ∈ Z/4Z, m ≥ 0}. (See Subsections 1.5 and 3.5, and notice that (A(1, 1)(1))H and A(1, 1) (resp. (sl(2, 2)(1))H and sl(2, 2)) are not the same; however they are closely related.) Define E i , E (m) ii−1, E (m) ii−1i−2 ∈ N+(⊂ G(E,Π, p)) (i ∈ Z/4Z,m ≥ 0) inductively by E i = Ei, E (m) i = [Ei, [Ei−1, E (m−1) i−2i−3i]], E (m) ii−1 = [Ei, E (m) i−1 ], E (m) ii−1i−2 = [Ei, E (m) i−1i−2]. Let Eii−1 = E (0) ii−1 and Eii−1i−2 = E (0) ii−1i−2. Let x : N+ → N− be the isomorphism such that x(Ei) = Fi (0 ≤ i ≤ 3). Denote (E,Π, p) by (AA) if its Dynkin diagram is the left one of (1) and if

In this paper, we construct an explicit basis of the quantized universal enveloping algebra Uq(slN + 1(C)). Let A = (%-)i