Abstract
In this crucial lecture we introduce the definition of the Lie algebra associated to a Lie group and its relation to that group. All three sections are logically necessary for what follows; §8.1 is essential. We use here a little more manifold theory: specifically, the differential of a map of manifolds is used in a fundamental way in §8.1, the notion of the tangent vector to an arc in a manifold is used in §8.2 and §8.3, and the notion of a vector field is introduced in an auxiliary capacity in §8.3. The Campbell-Hausdorff formula is introduced only to establish the First and Second Principles of §8.1 below; if you are willing to take those on faith the formula (and exercises dealing with it) can be skimmed. Exercises 8.27–8.29 give alternative descriptions of the Lie algebra associated to a Lie group, but can be skipped for now.
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© 2004 Springer Science+Business Media New York
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Fulton, W., Harris, J. (2004). Lie Algebras and Lie Groups. In: Representation Theory. Graduate Texts in Mathematics, vol 129. Springer, New York, NY. https://doi.org/10.1007/978-1-4612-0979-9_8
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DOI: https://doi.org/10.1007/978-1-4612-0979-9_8
Publisher Name: Springer, New York, NY
Print ISBN: 978-3-540-00539-1
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