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By Denis S. Arnon, Bruno Buchberger

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B) We first show that G acts irreducibly on Kn if and only if n = 1: Let G act irreducibly, and let A := K G ⊆ Kn×n be the (non-commutative) Ksubalgebra of Kn×n generated by G. Hence A acts faithfully on Kn , thus by Schur’s Lemma and the double centraliser theorem we have A = Kn×n . Since for all A, B ∈ G we have Tr((A − En )B) = Tr(AB) − Tr(B) = 0, we conclude Tr((A − En )C) = 0 for all C ∈ Kn×n , implying Tr(Eij (A − En )Ekl ) = 0 for all i, j, k, l ∈ {1, . . , n}, where Eij = [δik δjl ]kl ∈ Kn×n is the [i, j]-th matrix unit.

R ] ⊆ Λ, for some r ∈ N0 , and signs 1 r i ∈ {±1}, for i ∈ {1, . . , r}, such that H = Wλ1 · · · · · Wλr . Proof. We may assume that for any λ ∈ Λ there is λ ∈ Λ such that ϕλ = ιG ϕλ . r For any r ∈ N0 and α := [α1 , . . , αr ] ⊆ Λ let ϕα : i=1 Vαi → G : [x1 , . . , xr ] → x1 · · · · · xr . Letting Wα := Wα1 · · · · · Wαr = im(ϕα ) we conclude that Wα ⊆ G is closed and irreducible. For β := [β1 , . . , βs ] ⊆ Λ, where s ∈ N0 , let αβ := [α1 , . . , αr , β1 , . . , βs ] ⊆ Λ be the concatenation of α and β.

E. concatenation of maps, in L(G) transports r to DerK (K[G], K1 ): For f ∈ K[G] let µ∗ (f ) = i=1 gi ⊗ hi ∈ K[G] ⊗K K[G], for some r ∈ N and suitable fi , gi ∈ K[G]. Hence we deduce (γ · γ )(f ) = r r (γ ⊗ γ )( i=1 gi ⊗ hi ) = i=1 γ(gi )γ (hi ). r For all x ∈ G we have λ∗x−1 (f )(y) = f (xy) = µ∗ (f )([x, y]) = i=1 gi (x)hi (y), r for all y ∈ G, and hence λ∗x−1 (f ) = i=1 gi (x)hi . Thus γ(f )(x) = γλ∗x−1 (f ) = r r • i=1 gi (x)γ(hi ), and hence γ(f ) = i=1 gi · γ(hi ). This yields (γ γ ) (1)(f ) = r r γ γ (f )(1) = γ( i=1 gi · γ (hi ))(1) = i=1 γ(gi )γ (hi ) = (γ · γ )(f ).

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