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Representations of Lie Algebras by Anthony Henderson

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APPENDIX

Solutions to the exercises

Solutions for Chapter 2 exercises

Solution to exercises 2.3.1 We start from the Jacobi identity and substitute the given formulas, using the bilinearity and skew-symmetry of the Lie bracket to simplify the results:

Image

Since x1, x2, x3 are linearly independent, we can conclude that

Image

which forces a = 1, b = −1.

If you think about it further, this calculation suffices to ensure the existence of the Lie algebra described in the question with these values of a and b, namely with basis x1, x2, x3 whose Lie bracket satisfies ...

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