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

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 booklet might 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, team jewelry and different subject matters

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

**Example text**

Suppose ﬁrst that n = 2. Then G = a 2 × b 2 is a direct product of two cyclic groups of order 2. 4. 4). 4 where χa , χb = ±1. 4. 5). Then x t x = E and direct calculations show that the group G(H) of group-like elements in H consists of 8 elements e1 + ea + eb + eab ± E; e1 + ea − eb − eab ± 0 1 ; 1 0 e1 − ea + eb − eab ± −1 0 ; 0 1 e1 − ea − eb + eab ± 0 1 −1 . 0 Hence G(H) is isomorphic to the group consisting of matrices ±E, ± 0 1 , 1 0 ± −1 0 , 0 1 ± 0 1 −1 0 which is isomorphic to the group D4 .

G and k1 + k2 + k3 + k4 = n. k 43 7 SIMPLE COLOR LIE SUPERALGEBRAS The number of pairwise non isomorphic gradings can be evaluated as follows. The multiplication by g0 ∈ Z2 × Z2 does not change the homogeneous components, but causes 4 ﬁxed point free permutations of k1 , k2 , k3 , k4 . Thus we have at most 6 pairwise nonisomorphic elementary gradings and if we assign on of the numbers to a ﬁxed element of the group, we obtain 1-1 correspondence of the tuples and the gradings. For example, if we have one of ki zero, we assign 0 to c and then we have the 1-1 correspondence between the gradings and diﬀerent tuples of the form (e(k1 ) , a(k2 ) , b(k3 ) ) while if none of ki is zero we should consider all tuples (e(k1 ) , a(k2 ) , b(k3 ) , c(k4 ) ) where the value of k1 is the smallest possible.

3 we obtain Ω(f )−1 ⊗ Ω(f ) = θR. 4) we obtain |G| = θn. Thus θ = |G| n . 5) 1 G/ H 2 (G, k∗ ) G/ G∗ G/ G G/ 1 such that each projective representation of G can be lifted to an ordinary linear representation of G∗ . Here H 2 (G, k∗ ) is the second cohomology group of G with coeﬃcients in the multiplicative group k∗ . Corresponding representations of G and of G∗ are irreducible simultaneously. The ﬁeld k is algebraically closed. 4, Chapter X] the second cohomology group H 2 (G, k∗ ) is trivial if and only if the group G is cyclic.