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 could be of curiosity to graduate scholars and researchers operating within the idea of Lie and Jordan algebras and superalgebras and their representations, Hopf algebras, Poisson algebras, Quantum teams, staff earrings and different issues

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**Additional info for Algebras, Representations and Applications: Conference in Honour of Ivan Shestakov's 60th Birthday, August 26- September 1, 2007, Maresias, Brazil**

**Sample text**

7) where Ψ(d)e = χ(d)e, d ∈ D, with some one-dimensional character χ of D. 8. If d ∈ D then Ψ(d)em = χ(d)χ([d, gm ])em for any m. Proof. Since (G∗ ) ⊆ D for any d ∈ D we obtain −1 Ψ(d)em = Ψ(d)Ψ(gm )e = Ψ(gm )Ψ(d)Ψ([d−1 , gm ])e = Ψ(gm )Ψ(d)Ψ([d, gm ])e = χ(d)χ([d, gm ])Ψ(gm )e = χ(d)χ([d, gm ])em for any d ∈ D. 9. 7) ⎛ ⎞ 1 0 ... 0 ⎜0 χ([d, g2 ]) . . ⎟ 0 ⎟ Ψ(d) = χ(d) ⎜ ⎝. . . . . . . . . . . . . . ⎠ . 0 0 . . χ([d, gn ]) In particular tr Ψ(d) = χ(d) ( n m=1 χ([d, gm ])). ∗ Take an element g ∈ G \ D whose image in G∗ /D has order t > 1 dividing n.

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Gn ) deﬁne isomorphic elementary gradings if and only if there is a permutation ν of indexes 1, 2, . . , n and g0 ∈ G such that g0 gν(i) = gi , for all i = 1, 2, . . , n [7]. In the case G = Z2 × Z2 the grading by any tuple is thus isomorphic to the grading by a tuple of the form (e(k1 ) , a(k2 ) , b(k3 ) , c(k4 ) ) where g (k) = g, . . , g and k1 + k2 + k3 + k4 = n. k 43 7 SIMPLE COLOR LIE SUPERALGEBRAS The number of pairwise non isomorphic gradings can be evaluated as follows. The multiplication by g0 ∈ Z2 × Z2 does not change the homogeneous components, but causes 4 ﬁxed point free permutations of k1 , k2 , k3 , k4 .