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

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

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**Example text**

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. 10. Choose a system of representatives gj of cosets G∗ /D in such a way that gtj+r = g r gtj for j = 0, . . , nt − 2 and for r = 0, . . , t − 1. Then Ψ(g)etj+r = r = 0, . . , t − 2, etj+r+1 , t t χ(g )χ([g, gtj ]) etj , r = t − 1. Proof. We have Ψ(g)etj+r = Ψ(g)Ψ(gtj+r )e = Ψ(ggtj+r )e. If r = 0, . . , t − 2, then ggtj+r = gtj+r+1 and therefore Ψ(g)etj+r = Ψ(gtj+r+1 )e = etj+r+1 .

U. Umirbaev, ”Free Akivis Algebras, primitive elements and hyperalgebras”, J. Algebra 250(2), (2002) 533-548. , ” The theory of Lie superalgebras”, LNM 716 Springer Verlag, Berlin 1979. A. Chubarov To Prof. P. Shestakov on the occasion of his 60th anniversary Abstract. The paper considers properties of two classes of ﬁnite dimensional semisimple Hopf algebras from [A]. We show that for any positive integer n > 1 there exists a semisimple Hopf algebra of dimension 2n2 from [A]. Introduction One of the most important problems in the theory of Hopf algebras is a classiﬁcation of ﬁnite dimensional semisimple Hopf algebras.

Scheunert’s (often called Scheunert’s trick, however see [17]) enabling one to pass from (G, β)-Lie superalgebras to ordinary Lie superalgebras with a number of properties preserved. 3) with the commutation factor β = βδ where δ(x, y) = σ(x, y)/σ(y, x). e. ε0 (g, h) = 1 except ε0 (g, h) = −1 for g, h ∈ G . 1. [19] Let G be a ﬁnitely generated abelian group, β : G×G → F ∗ a commutation factor with G = G+ . Then there exists a 2-cocycle σ ∈ Z 2 (G, F ∗ ) such that βδ = 1. ). Since βε0 satisﬁes the conditions of the theorem we can ﬁnd σ with βε0 δ = 1 whence βδ = β0−1 = ε0 .