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 publication might 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, workforce earrings and different subject matters
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Extra resources for Algebras, Representations and Applications: Conference in Honour of Ivan Shestakov's 60th Birthday, August 26- September 1, 2007, Maresias, Brazil
3) that ⎞ ⎛ Ω(g −1 ) ⊗ Ω(g)⎠ (E ⊗ Ω(h)) = ⎝ g∈G g∈G µg,h Ω(g = g∈G Ω(g −1 ) ⊗ Ω(g)Ω(h) −1 −1 µg,h µ−1 ⊗ Ω(gh) g,h Ω(h)Ω(gh) ) ⊗ Ω(gh) = ⎛ = (Ω(h) ⊗ E) ⎝ g∈G g∈G ⎞ Ω(gh)−1 ⊗ Ω(gh)⎠ ⎛ ⎞ = (Ω(h) ⊗ E) ⎝ Ω(f )−1 ⊗ Ω(f )⎠ f ∈G PROPERTIES OF SOME SEMISIMPLE HOPF ALGEBRAS 29 7 Since the representation Ω is irreducible the linear span of all Ω(h), h ∈ G, coincides with Mat(n, k). Thus ⎛ ⎞ ⎛ ⎞ (A ⊗ E) ⎝ Ω(f )−1 ⊗ Ω(f )⎠ = ⎝ f ∈G f ∈G for any matrix A. Similarly ⎛ ⎞ ⎛ Ω(f )−1 ⊗ Ω(f )⎠ = ⎝ (E ⊗ B) ⎝ Ω(f )−1 ⊗ Ω(f )⎠ (E ⊗ A) f ∈G ⎞ Ω(f )−1 ⊗ Ω(f )⎠ (B ⊗ E) f ∈G for any B.
K < s < r: < er , es , ek >= ¯ (−1)s¯r¯(es er ) ∗ ek + [er , es ] ∗ ek − (−1)s¯k er ∗ (ek es ) − er ∗ [es , ek ] = ¯ r (¯ s+k)¯ s¯r¯ [es , ek ] ∗ er + (−1) es ∗ [er , ek ] + [er , es ] ∗ ek − er ∗ [es , ek ]− = (−1) ¯ ¯ −(−1)(¯s+¯r)k ek ∗ [er , es ] − (−1)s¯k [er , ek ] ∗ es + (−1)s¯r¯A(es , er , ek )− ¯ ¯ ¯ ¯ −(−1)(¯s+k)¯r A(es , ek , er ) + (−1)s¯k+(¯s+k)¯r A(ek , es , er ) − (−1)(¯s+¯r)k A(ek , er , es )+ ¯ +(−1)s¯k A(er , ek , es ) = ( since [x, y] =< x, y >, all x, y ∈ M ) ¯ = SJ(er , es , ek ) − ((−1)s¯r¯A(es , er , ek ) − (−1)(¯s+k)¯r A(es , ek , er )+ ¯ ¯ ¯ ¯ +(−1)s¯k+(¯s+k)¯r A(ek , es , er ) − (−1)(¯s+¯r)k A(ek , er , es ) + (−1)s¯k A(er , ek , es )) = = A(er , es , ek ) ( by the deﬁnition of Akivis superalgebra).
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. 1) i = ([b, a]a) = (c−1 a)i = ai c−i i Consider a linear representation Ψ of the group G∗ of dimension n in a space V induced by a one-dimensional representation Ψ of b × c in one-dimensional space W with one basic element e such that Ψ(b)e = ωe, Ψ(c)e = ηe.
Algebras, Representations and Applications: Conference in Honour of Ivan Shestakov's 60th Birthday, August 26- September 1, 2007, Maresias, Brazil by Vyacheslav Futorny, Victor Kac, Iryna Kashuba, Efim Zelmanov