By Vyacheslav Futorny, Victor Kac, Iryna Kashuba, Efim Zelmanov

This quantity comprises contributions from the convention on 'Algebras, Representations and functions' (Maresias, Brazil, August 26 - September 1, 2007), in honor of Ivan Shestakov's sixtieth birthday. This e-book may be of curiosity to graduate scholars and researchers operating within the concept of Lie and Jordan algebras and superalgebras and their representations, Hopf algebras, Poisson algebras, Quantum teams, workforce jewelry and different themes

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More precisely an = bn = cn = 1, [a, b] = aba−1 b−1 = c, [a, c] = [b, c] = 1. 1) i = ([b, a]a) = (c−1 a)i = ai c−i i Consider a linear representation Ψ of the group G∗ of dimension n in a space V induced by a one-dimensional representation Ψ of b × c in one-dimensional space W with one basic element e such that Ψ(b)e = ωe, Ψ(c)e = ηe. Here ω, η are primitive roots of 1 of degree n. 7) in V has the form e1 = e, e2 = Ψ(a)e, . . , en = Ψ(a)n−1 e. 2) Ψ(c)ei = ηei . 3) Ep+i,p ω j η −j(p−1) ∈ GL(n, k), Ψ(ai bj ) = Ψ(cl ) = η l E p∈Zn for 0 i, j n − 1.

4). 2. 1). Proof. This is a simple computation. Take homogeneous a, b ∈ A and compare the β -bracket in Aσ and the σ-twisted β-bracket in [A]. That is, [a, b]β proving our claim. = = = = = (ab)σ − β (a, b)(ba)σ σ(a, b)ab − β (a, b)σ(b, a)ba σ(a, b)(ab − (β δ −1 )(a, b)ba) σ(a, b)(ab − β(a, b)ba) σ(a, b)[a, b]β = ([a, b]β )σ SIMPLE COLOR LIE SUPERALGEBRAS 39 3 The second part of this theorem gives yet another way of replacing the study of color Lie superalgebras by the study of the ordinary ones.

G| g∈G Proof. Suppose that the projective representation Ω is reducible. Then there exists an invertible matrix P ∈ GL(n, k) such that P Ω(g)P −1 = Bg 0 0 Cg with square blocks Bg , Cg of a smaller size. 2). Conversely let a representation Ω be irreducible. 3) −1 Ω(g)−1 = µ−1 g,h Ω(h)Ω(gh) where µg,h ∈ k∗ . 3) that ⎞ ⎛ Ω(g −1 ) ⊗ Ω(g)⎠ (E ⊗ Ω(h)) = ⎝ g∈G g∈G µg,h Ω(g = g∈G Ω(g −1 ) ⊗ Ω(g)Ω(h) −1 −1 µg,h µ−1 ⊗ Ω(gh) g,h Ω(h)Ω(gh) ) ⊗ Ω(gh) = ⎛ = (Ω(h) ⊗ E) ⎝ g∈G g∈G ⎞ Ω(gh)−1 ⊗ Ω(gh)⎠ ⎛ ⎞ = (Ω(h) ⊗ E) ⎝ Ω(f )−1 ⊗ Ω(f )⎠ f ∈G PROPERTIES OF SOME SEMISIMPLE HOPF ALGEBRAS 29 7 Since the representation Ω is irreducible the linear span of all Ω(h), h ∈ G, coincides with Mat(n, k).