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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It is a well-known result that the fixed point subalgebra of a finite dimensional complex simple Lie algebra under a finite order automorphism is a reductive Lie algebra so it is a direct sum of finite dimensional simple Lie subalgebras and an abelian subalgebra. We consider this for the class of extended affine Lie algebras and are able to show that the fixed point subalgebra of an extended affine Lie algebra under a finite order automorphism (which satisfies certain natural properties) is a sum of extended affine Lie algebras (up to existence of some isolated root spaces), a subspace of the center and a subspace which is contained in the centralizer of the core. Moreover, we show that the core of the fixed point subalgebra modulo its center is isomorphic to the direct sum of the cores modulo centers of the involved summands.

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