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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 + (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.

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 + (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.

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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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 + (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.
Transcribed Image Text: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 + (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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