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

This quantity comprises 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 booklet might 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, team jewelry and different issues

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**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**

4) Eij ⊗ Eji = R= i,j 1 n i, j n, stand for matrix units in U t Ag−1 ⊗ t Ag U −1 . 4) that Ag−1 = A−1 g for all g ∈ G and A1 = E. 3. 5) J= 0 E −E 0 where E is the unit matrix of the size m. 5) are canonical forms of skew-symmetric matrices. 1). 2 is a group-like element if and only if the following conditions are satisﬁed: 1) χgh = χg χh for all g, h ∈ G; 2) g x = χg x = x g for any g ∈ G; 3) xU t x = U . 2). 2 is cocommutative if and only if g x=x g for all g ∈ G and x ∈ Mat(n, k). 6. 2 is cocommutative if and only if Ag = ξg U t Ag U −1 for all g ∈ G and x ∈ Mat(n, k) where ξg = ±1.

It is known that G ∼ = G, so G is also elementary. Next we set H = Λ⊥ = {g ∈ G | λ(g) = 1 ∀λ ∈ Λ}. Similarly we deﬁne K = Π⊥ . Now we deﬁne an K-grading on M by saying deg x = k if λ ∗ x = λ(k)x. Now we consider L = F [H] ⊗ M as described in the theorem, with the operation and the grading deﬁned therein. 1) (π(h))−1 (π ∗ x). ϕ(h ⊗ x) = π∈Π SIMPLE COLOR LIE SUPERALGEBRAS 41 5 We have to check that ϕ is an isomorphism of graded algebras. First of all, it is easy to check that L = π∈Π (π ∗ M ). Then ϕ([h ⊗ x, h ⊗ y]) (π(h))−1 (π ∗ x), = [ π∈Π ρ∈Π [(π(h)) = (ρ(h ))−1 (ρ ∗ y)] −1 (π ∗ x), (π(h ))−1 (π ∗ y)] π∈Π (π(hh ))−1 (π ∗ [x, y] = ϕ((hh ) ⊗ [x, y]).

12. 3) in PGL(n, k) we have 1 = [Ag , U t Ah U −1 ] = [LΨ(g)L−1 , U t L−1 t Ψ(h) t LU −1 ] = L[Ψ(g), L−1 U t L−1 t Ψ(h) t LU −1 L]L−1 . Put Λ = L−1 U t L−1 ∈ GL(n, k). A. 13. 2). 2. 2 in which U = E is the identity matrix and G is a direct product G = a × b of two cyclic groups a , b of order n. 5 the derived subgroup [G∗ , G∗ ] = c is a central cyclic group of order n. 5 the group G∗ is a semidirect product of a normal subgroup b × c by a cyclic subgroup a . More precisely an = bn = cn = 1, [a, b] = aba−1 b−1 = c, [a, c] = [b, c] = 1.