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 ebook may 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, crew jewelry and different themes

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**Sample text**

To obtain all gradings up to isomorphism one has to apply to the representatives of the classes of weak equivalence the automorphisms of Z2 ×Z2 or, which is the same, all possible permutations of nontrivial elements, six in total. We will see, however that in the whole number of cases there is no need to apply the group automorphisms since some of them produce the same result as the algebra automorphisms and the respective algebras are not just weakly equivalent but isomorphic as graded algebras.

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. Kac Hopf algebra H8 [KP].

If G is cyclic then G∗ = G because H 2 (G, k∗ ) is trivial. Then all irreducible representations of G have dimension 1. 4 the representation Ψ is irreducible and has dimension n. Now we have the following situation. 5. 6. 1] which means that it is induced by a one-dimensional representation of some subgroup D ⊂ G∗ . Clearly D contains the derived central subgroup [G∗ , G∗ ] and therefore D is normal in G∗ . Consider an irreducible linear representation Ψ of G∗ in a space V of dimension n induced by a one-dimensional representation of a normal subgroup D, containing [G∗ , G∗ ].