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

ISBN-10: 0821846523

ISBN-13: 9780821846520

This quantity includes 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 publication should 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, crew jewelry and different themes

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**Example text**

1) is hk. To ﬁnd the degree of the right side we have to apply λρ. Then we obtain (π(h))−1 (ρπ ∗ (λ ∗ x)) (λρ) ∗ (ϕ(h ⊗ x)) = π∈Π = λ(k)ρ(h) ((πρ)(h))−1((ρπ) ∗ x) π∈Π = (λρ)(hk) (σ(h))−1 (σ ∗ x), σ∈Π as needed. This proposition allows us to conclude that in the case of Z2 ×Z2 -simple graded algebras we have to consider three cases (in every one g is a simple Lie algebra) (1) L = F [Z2 × Z2 ] ⊗ g, with no grading on M ; (2) L = F [Z2 ] ⊗ g, with a Z2 -grading on g; (3) L = g, with a Z2 × Z2 -grading on g.

Thus χa = χb = ±1. By direct calculation using the equation xU x = U we can ﬁnally obtain that the group G(H) consists of 4 elements t e1 + ea + eb + aab ± E, e1 − ea − eb + aab ± 0 1 . 1 0 So this case corresponds to unique Hopf algebra non-isomorphic to group algebras of D4 and Q8 . Hence H H8 . 6 is not satisﬁed and H is not cocommutative. Finally it is necessary to mention that S. 13. 2 by a (skew-)symmetric matrix U and by an irreducible projective representation g → Ag of the group G. 2] H is uniquely determined by the representation up to an orthogonal equivalence for U = E and up to a symplectic equivalence if U = S of representations.

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.

### Algebras, Representations and Applications: Conference in Honour of Ivan Shestakov's 60th Birthday, August 26- September 1, 2007, Maresias, Brazil by Vyacheslav Futorny, Victor Kac, Iryna Kashuba, Efim Zelmanov

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