Abstract
Let g be a reductive Lie algebra over R, G a connected Lie group with Lie algebra g. In the hands of Harish-Chandra (and others) the universal enveloping algebra G of gc (the complexification of g) plays a prominent role in the representation theory of G. This will become apparent in Chapter 4 where we shall see, for instance, how a representation of G on a Banach space E, say, gives rise in a natural manner to various linear representations of G on subspaces of E. Moreover the study of these representations of G yields important information about the given representation of G. This chapter, then, which is algebraic in character, deals on the one hand with the structure of G and, on the other, with various aspects of its representation theory. [Actually it will be just as easy to carry out most of the discussion in the context of a reductive Lie algebra g over an algebraically closed field k of characteristic zero — unless specifically stated to the contrary, this will be the underlying assumption in what follows.]
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© 1972 Springer-Verlag Berlin Heidelberg
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Warner, G. (1972). The Universal Enveloping Algebra of a Semi-Simple Lie Algebra. In: Harmonic Analysis on Semi-Simple Lie Groups I. Die Grundlehren der mathematischen Wissenschaften in Einzeldarstellungen mit besonderer Berücksichtigung der Anwendungsgebiete, vol 188. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-50275-0_2
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DOI: https://doi.org/10.1007/978-3-642-50275-0_2
Publisher Name: Springer, Berlin, Heidelberg
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