Question: Lie Algebra In this question, we consider invariant bilinear forms. (a) Given an L-module V and a bilinear form (., .) on V, prove that

Lie Algebra

In this question, we consider invariant bilinear forms. (a) Given an L-module V and a bilinear form (., .) on V, prove that (., . ) : V - V* V H (v, .) defines an L-module homomorphism from V to V* if and only if (., .) is invariant. (b) Let L be a finite-dimensional simple Lie algebra. Prove that any invariant bilin- ear form on L is a scalar multiple of the Killing form

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