Titlebook: Compact Lie Groups; Mark R. Sepanski Textbook 2007 Springer-Verlag New York 2007 Group theory.Lie algebra.Representation theory.algebra.ca

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Lie Algebras,ce to the identity. The resulting object is called a Lie algebra. Simply by virtue of the fact that vector spaces are simpler than groups, the Lie algebra provides a powerful tool for studying Lie groups and their representations.

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Highest Weight Theory, Two important problems remain. The first is to parametrize ? in a reasonable manner and the second is to individually construct each irreducible representation in a natural way. The solution to both of these problems is closely tied to the notion of . weights.

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https://doi.org/10.1007/978-3-322-94211-1By examining the joint eigenvalues of a Cartan subalgebra under the ad-action, a great deal of information about a Lie group and its Lie algebra may be encoded. For instance, the fundamental group can be read off from this data (§ 6.3.3). Moreover, this encoding is a key step in the classification of irreducible representations (§7).

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Representations,Lie groups are often the abstract embodiment of symmetry. However, most frequently they manifest themselves through an action on a vector space which will be called a representation. In this chapter we confine ourselves to the study of finite-dimensional representations.