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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).

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