Advanced Topics LIE ALGEBRAS (Date Due: 30thJuly) ASSIGNMENT 2 Question 1. Let L be the real vector space R3. Given x; y 2 L dene [x; y] := x y; where denotes the usual cross product of vectors. Sow that L is a Lie algebra and determine its structure constants relative to the standard basis for R3. Question 2. Let be a derivation of the Lie algebra L. Show that if commutes with every inner derivation then (L) C(L); where C(L) denotes the centre of L. Question 3. Let x 2 gl(n; F) have n distinct eigenvalues 1; 2; ; n in F. Prove that the eigenvalues of adx are the n2 scalars i ?? j ; (1 i; j n): (Note that only n2 ?? n + 1 scalars are paiwise distinct from each other since i ?? i = 0 for all i.) 1
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