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

ISBN-10: 0821846523

ISBN-13: 9780821846520

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 e-book could 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 themes

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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**

**Sample text**

Since (G∗ ) ⊆ D for any d ∈ D we obtain −1 Ψ(d)em = Ψ(d)Ψ(gm )e = Ψ(gm )Ψ(d)Ψ([d−1 , gm ])e = Ψ(gm )Ψ(d)Ψ([d, gm ])e = χ(d)χ([d, gm ])Ψ(gm )e = χ(d)χ([d, gm ])em for any d ∈ D. 9. 7) ⎛ ⎞ 1 0 ... 0 ⎜0 χ([d, g2 ]) . . ⎟ 0 ⎟ Ψ(d) = χ(d) ⎜ ⎝. . . . . . . . . . . . . . ⎠ . 0 0 . . χ([d, gn ]) In particular tr Ψ(d) = χ(d) ( n m=1 χ([d, gm ])). ∗ Take an element g ∈ G \ D whose image in G∗ /D has order t > 1 dividing n. 10. Choose a system of representatives gj of cosets G∗ /D in such a way that gtj+r = g r gtj for j = 0, .

Hence its inverse coincides with its transpose and therefore they commute. 2). Hence its transpose coincides with itself. It follows that the transpose of Ψ(ai bj ) is equal to Ψ(bj a−i ) in GL(n, k) and to Ψ(a−i bj ) in PGL(n, k). 1) matrices Ψ(a−i bj ), Ψ(ar bs ) commute in PGL(n, k) and the ﬁrst statement is proved. 33 11 PROPERTIES OF SOME SEMISIMPLE HOPF ALGEBRAS It is interesting to mention that the same projective representation of the same group G = a × b is used in [BSZ] for the classiﬁcation of group gradings on full matrix algebras Mat(n, k) (by an Abelian group).

En = Ψ(a)n−1 e. 2) Ψ(c)ei = ηei . 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 ).

### 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

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