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

This quantity includes 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 e-book can be of curiosity to graduate scholars and researchers operating within the concept of Lie and Jordan algebras and superalgebras and their representations, Hopf algebras, Poisson algebras, Quantum teams, team jewelry and different themes

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3) Ep+i,p ω j η −j(p−1) ∈ GL(n, k), Ψ(ai bj ) = Ψ(cl ) = η l E p∈Zn for 0 i, j n − 1. 1. 13 with Λ = E. Proof. 4] we need to show that for any ai , 1 i n − 1 there exists an element y ∈ H such that χ(y) = χ(ai ya−i ) or χ([ai , y]) = 1. Taking y = b we obtain χ([ai , b]) = χ([a, b])i = χ(c)i = η i = 1 because η is a primitive root of 1 of degree n. 4 we need to show that each matrix Ψ(ai bj ) commutes in PGL(n, k) with the transpose of another matrix Ψ(ar bs ). Each matrix Ψ(aj ) is a permutation matrix.

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 .

Then, in gr U (M ) there holds ˜1 (M ) = ι([x, y]) + U ˜1 (M ) = 0; x ¯y¯ − (−1)αβ y¯x ¯ = (ι(x)ι(y) − (−1)αβ ι(y)ι(x)) + U ˜2 (M ) = 0. ˜ 2(M ) = ι(A(x, y, z))+ U (¯ xy¯)¯ z −x ¯(¯ y z¯) = (ι(x)ι(y))ι(z)−ι(x)(ι(y)ι(z))+ U ∗ This implies that I ⊆ Ker τ . Therefore, there is an epimorphism of Z-graded ˜ (M ) such that τ˜(m + I ∗ ) = τ (m) = ι(m), for all superalgebras τ˜ : T˜(M )/I ∗ → gr U m ∈ M. From these two lemmas, we know that the composite map ˜ (M ) τ˜π ˜ −1 : V (M ) → gr U is an epimorphism of superalgebras satisfying τ˜π ˜ −1 (m) = ι(m) for all m ∈ M .