# Berkeley MATH 261A - MATH 261A Exercises (3 pages)

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# MATH 261A Exercises

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## MATH 261A Exercises

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Pages:
3
School:
University of California, Berkeley
Course:
Math 261a - Lie Groups
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Exercise 125 Show that for the Lie algebra gln k with ei the matriix with just one non zero entry a 1 in position i on the diagonal we have ei ej 2n 2 if i j and 2 if i 6 j Deduce that the Killing form on sln k is 2n times the symmetric bilinear form associated to the standard representation but the Killing form on gln R is not a multiple of the form of the standard representation and has a non trivial kernel Exercise 126 If k is a field of characteristic 2 then the semidirect product gl2 k k 2 is solvable The Killing form is not identically zero on its derived algebra sl2 k k 2 Exercise 127 Let G be the Lie algebra slp Fp of 2 by 2 matrices over the field of p elements Show that G is simple if p 2 Show that the Killing form of G is identically 0 Show that Trace AB is a non degenerate invariant bilinear form on G Exercise 128 Suppose G is the Lie algebra over a field of characteristic p 0 with basis ai for i Z pZ and bracket ai aj i j ai j Show that G is a simple Lie algebra but has no non zero invariant bilinear form Lemma 129 Dieudonne Suppose that a finite dimensional Lie algebra over any field of any characteristic has a non degenerate bilinear form and no abelian ideals Then it is a direct sum of simple subalgebras Proof Fix a minimal ideal M The derived ideal M M is contained in M and cannot be 0 as M is non abelian so M M M is perfect The orthogonal complement N of M is also an ideal as the bilinear form P is invariant P It cannot contain M as otherwise we would have x m x ai bi x ai bi 0 so M would be in the kernel of which is not possible So N M 0 as it is a proper ideal of the minimal ideal M So G splits as the direct sum of M and N so M is simple and continuing by induction so is N Exercise 130 Show that if L is a Lie algebra with an invariant symmetric bilinear form then L t tn has an invariant symmetric bilinear form given by the coefficient of tn 1 of the bilinear form on L t tn with values that are truncated power series If the form on L is non

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