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

Abstract

Let U (χ) be a generalized quantum group 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'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 R(χ) := R + (χ) ∪ −R + (χ).(1) Simple Lie algebras of type X N , where X = A, . . . , G, (2) sl(m + 1|n + 1) (m + n ≥ 2),The ones in (1) and (3) are simple. The simple Lie superalgebras A(m, n) are defined by sl(m + 1|n + 1) if m = n, and otherwise A(n, n) := sl(n + 1|n + 1)/i, where i is its unique one-dimensional ideal.Bases of the root systems of the Lie superalgebras of (2)-(3) are not conjugate under the action of their Weyl groups. However each two of them are transformed to each other under the action of their Weyl groupoids W , whose axiomatic treatment was introduced by Heckenberger and the second author [12]. Kac [16, Theorem 8 (c)] gave a list of finite-dimensional irreducible highest weight modules of the Lie superalgebras in (2)-(3) above. In the same way as in this paper, we can have a new proof of recovering the list; our idea is to use a specially good one among the reduced expressions of the longest element (with a 'standard' end domain) of the Weyl groupoid W , see also Remark 14.Let g := sl(m + 1|n + 1) or C(n) for example. Let h be a Cartan subalgebra of g such that the Dynkin diagram of (g, h) is a standard one. Let Π = {α i |1 ≤ i ≤ dim h} be the set of simple roots α i corresponding to h. Let w 0 be the longest element of W of g whose end domain is corresponding to h. Then the length ℓ(w 0 ) of w 0 is equal to the number of positive roots of g. Let k be the the number of even positive roots of g. The key fact used in this paper is that there exists a reduced expression s i 1 · · · s i ℓ(w 0 ) of w 0 such that s i 1 · · · s i x−1 (α ix ), 1 ≤ x ≤ k, are even positive roots, and s i 1 · · · s i y−1 (α iy ), k + 1 ≤ y ≤ ℓ(w 0 ), are odd positive roots. We claim that this is essential to the fact that an irreducible highest weight g-module of highest weight Λ is finite-dimensional if and only if 2 Λ,α i α i ,α i ∈ Z ≥0 for all even simple roots α i , where , is the bilinear form coming from the Killing form of g.Motivated by Andruskiewitsch and Schneider's theory [3], [4] toward the classification of pointed Hopf algebras, Heckenberger [9] classified the Nichols algebras of diagonal-type. Let K be a characteristic zero field. Let U(χ) be the K-algebra defined in the same manner as in the Lusztig's book [18, 3.1.1 (a)-(e)] for any bi-homomorphism χ : ZΠ × ZΠ → K × , where Π = {α i |i ∈ I} is the set of simple roots of the Kharchencko's positive root system R + (χ) associated with χ. We call U(χ) the generalized quantum group. We say that χ (or U(χ)) is of finite-type if R + (χ) is finite and irreducible. We say that χ (or U(χ)) is of finite-and-infinite-dimensional-type (FID-type, for short) if χ is o...

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Representations of quantum affine superalgebras

“…Note that early in [44], Yamane generalised Beck's argument [3] by using the Weyl groupoids instead of the Weyl groups to get the two presentations of U q (Lsl(M, N )). Later in [20], similar arguments of Weyl groupoids led to Drinfel'd realizations of the quantum affine superalgebras U q (D (1) ( 2, 1; x)). Also, in the paper [42], Serganova studied highest weight representations of certain classes of Kac-Moody superalgebras with the help of Weyl groupoids.…”

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