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 ebook can be of curiosity to graduate scholars and researchers operating within the idea of Lie and Jordan algebras and superalgebras and their representations, Hopf algebras, Poisson algebras, Quantum teams, crew earrings 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**

**Example text**

Research is partially supported by grants RFFI 06-01-00037. Research is partially supported by grants RFFI 06-01-00037. A. 1]. 1) where {eg , g ∈ G, E} is a set of central indecomposable idempotents and E is the identity matrix. Since dim H = |G| + n2 the order of G is a divisor of n2 . Recall that there are left and right actions f x, x f of a dual Hopf algebra H ∗ on H, namely if f ∈ H ∗ , x ∈ H, and ∆(x) = x x(1) ⊗ x(2) then f x= x(1) f, x(2) , x f= f, x(1) (x(2) ). x x The convolution multiplication f ∗ g in H ∗ has the form f ∗ g, x = µ(f ⊗ g)∆(x) = f, x(1) g, x(2) x = f, g x = g, x f , ∗ where x ∈ H, f, g ∈ H and µ : H ⊗ H → H – is the multiplication map in H.

For n ≥ 1, the map µn : V n (M ) → (gr V˜ (M ))n deﬁned by µn (v) = v + V˜n−1 (M ) is an isomorphism of vector spaces. So taking µ0 = IdK , µ = ⊕n≥0 µn : V (M ) → gr V˜ (M ) is an isomorphism of Z-graded vector spaces. Looking at the formulas which deﬁne the multiplication in V˜ (M ), it is easy to see that this is in fact an isomorphism of Z-graded su˜ (M ) → gr V˜ (M ) → V (M ) peralgebras. The composite homomorphisms µ−1 ˜ : gr U −1 ∗ ˜ ˜ : V (M ) → T (M )/I → gr U (M ) (recall the preceeding lemmas) are and τ˜π ˜ inverse of each other.

2 that the superalgebras arising in diﬀerent case are mutually non-isomorphic. 5. (Z2 × Z2 , β)-superalgebra structures on classical simple algebras By deﬁnition, given two G-gradings g = ⊕g∈G gg and g = ⊕g∈G gg of an algebra g by a group G, we call them equivalent or isomorphic if there exists an automorphism α of g such that gg = α(gg ). We also say that in this case two graded algebras in question are isomorphic. In [9] the classiﬁcation of gradings was presented with the use another equivalence relation on the gradings.