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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.

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