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 publication can be of curiosity to graduate scholars and researchers operating within the conception of Lie and Jordan algebras and superalgebras and their representations, Hopf algebras, Poisson algebras, Quantum teams, staff jewelry and different subject matters

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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 idea of Lie and Jordan algebras and superalgebras and their representations, Hopf algebras, Poisson algebras, Quantum teams, workforce jewelry and different themes

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

**Sample text**

And U. 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.

4) and tr Ag = nδg,1 . 4) is equivalent to the irreducibility of the representation. For an irreducible representation the equality tr Ag = nδg,1 holds. 4) and tr Ag = nδg,1 . 2 with U = E. It is necessary to mention some other paper considering the same class of Hopf algebras. In the paper [T] there is given an explicit form of H if the order of G is n2 and either n is odd or the group G is an elementary Abelian 2-group. In the paper [TY] it is shown that if n = 2 then there exist up to equivalence four classes of Hopf algebras H, namely group algebras of Abelian groups of order 8, the group algebras of the dihedral group D4 , of the quaternions Q8 , and G.

5) and xU t x = U . Moreover χg x = U t Ψ(g)U −1 xU t Ψ(g)−1 U −1 , g = a, b ∈ G. Again direct calculations show that in this case the group G(H) of group-like elements in H consists of 8 elements e1 + ea + eb + eab ± E; e1 + ea − eb − eab ± 0 i i , 0 i2 = −1; 35 13 PROPERTIES OF SOME SEMISIMPLE HOPF ALGEBRAS e1 − ea + eb − eab ± −i 0 e1 − ea − eb + eab ± 0 1 0 , i i2 = −1; −1 . 0 Hence G(H) is isomorphic to the group consisting of matrices ±E, ± 0 i , i 0 ± −i 0 , 0 i ± 0 1 −1 0 which is isomorphic to the quaternion group Q8 and H is the group algebra of Q8 .