Lie Algebras [Lecture notes] by Fulton B. Gonzalez

By Fulton B. Gonzalez

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So, amusingly, if the center c of g is {0}, we have an increasing chain of Lie algebras g ⊂ Der g ⊂ Der (Der g) ⊂ Der (Der (Der g)) ⊂ · · · According to a theorem by Schenkmann, this chain eventually stops growing. 6. Let a and b be ideals of g such that g = a ⊕ b. We say that g is a direct sum of the ideals a and b. 7. If c = {0}, then show that g is complete if and only if g is a direct summand of any Lie algebra m which contains g as an ideal: m = g ⊕ h, where h is another ideal of m. 8. If g is simple, show that Der g is complete.

3) are immediately satisfied. Thus, any vector space may be endowed with the (obviously trivial) structure of an abelian Lie algebra. 5. In R3 , show that a × (b × c) = (a · c)b − (a · b)c. Then show that the cross product is a Lie bracket on the real vector space R3 , so that R3 is a 3-dimensional Lie algebra over R. 6. Here’s a great source of Lie algebras: take any associative algebra A, and define a Lie bracket on A by putting [x, y] := xy − yx Of course we need to verify that [x, y] is indeed a Lie bracket on A.

Then S ′ and N ′ commute with T , and so must commute with both S and N . 11. 13. The only eigenvalue of S − S ′ is therefore 0, whence S − S ′ = 0. Therefore, N ′ − N = 0. 10 Symmetric Bilinear Forms Let V be a vector space over F. 30) (v, w) → v, w which is linear in each of its two arguments: αv + βv ′ , w = α v, w + β v ′ , w v, αw + βw′ = α v, w + β v, w′ , for all v, v ′ , w, w′ ∈ V and all α, β ∈ F. 1. The dot product on Rn is a bilinear form. More generally, an inner product on a real vector space V is a bilinear form.

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