Titlebook: Elements of the Theory of Representations; Aleksandr A. Kirillov Book 1976 Springer-Verlag Berlin Heidelberg 1976 Darstellung.Group repres

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The Method of Orbitsclosely connected with a certain special finite-dimensional representation of this group. This representation acts in the dual space {{g}}* of the Lie algebra {{g}} of the group under study. We will call it a [[co-adjoint]] or briefly a [[K-representation]]

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https://doi.org/10.1007/978-1-349-02154-3Proofs of the facts given in this section, along with more information, can be found in the textbook of S. Lang [39] and also in the treatise of N. Bourbaki [6].

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https://doi.org/10.1007/978-981-13-3372-9Sets with structure locally like Euclidean spaces are called manifolds. This property enables us to introduce local systems of coordinates on manifolds and to employ the apparatus of mathematical analysis. A precise definition of manifold follows.

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https://doi.org/10.1007/978-3-031-57683-6A set G is called a Lie group if it is a topological group and a smooth manifold for which the mapping ., given by φ(.)=. is smooth.

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Jaspreet Kaur,Manishi Mukesh,Akshay AnandWe have already stated . that the term “representation” in the wide sense means a homomorphism of the group . into the group of one-to-one mappings of a certain set . onto itself.,A representation . is called . if . is a linear space and the mappings . are linear operators.

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https://doi.org/10.1007/978-3-319-07944-8One of the principal problems of the theory of representations is the problem of decomposing representations of a group . into the simplest possible components.

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