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Pivych
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I need to solve two assignments in Lie algebras. These assignments are not very difficult, but my knowledges in Lie algebras aren't good.
1. Let [tex]\delta[/tex] be a derivation of the Lie algebra [tex]\Im[/tex]. Show that if [tex]\delta[/tex] commutes with every inner derivation, then [tex]\delta[/tex]([tex]\Im[/tex])[tex]\subseteq[/tex]C([tex]\Im[/tex]), where C([tex]\Im[/tex]) denotes the centre of [tex]\Im[/tex] .
2. Let x [tex]\in[/tex] gl(n,F) have n distinct eigenvalues [tex]\lambda[/tex]1..[tex]\lambda[/tex]n in F. Prove that eigenvalues of ad[tex]_{}x[/tex] are the n[tex]^{}2[/tex] scalars [tex]\lambda[/tex][tex]_{}i[/tex]-[tex]\lambda[/tex][tex]_{}j[/tex] (1[tex]\leq[/tex]i,j[tex]\leq[/tex]n)
Your prompt reply will be highly appreciated
1. Let [tex]\delta[/tex] be a derivation of the Lie algebra [tex]\Im[/tex]. Show that if [tex]\delta[/tex] commutes with every inner derivation, then [tex]\delta[/tex]([tex]\Im[/tex])[tex]\subseteq[/tex]C([tex]\Im[/tex]), where C([tex]\Im[/tex]) denotes the centre of [tex]\Im[/tex] .
2. Let x [tex]\in[/tex] gl(n,F) have n distinct eigenvalues [tex]\lambda[/tex]1..[tex]\lambda[/tex]n in F. Prove that eigenvalues of ad[tex]_{}x[/tex] are the n[tex]^{}2[/tex] scalars [tex]\lambda[/tex][tex]_{}i[/tex]-[tex]\lambda[/tex][tex]_{}j[/tex] (1[tex]\leq[/tex]i,j[tex]\leq[/tex]n)
Your prompt reply will be highly appreciated
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