The content of this book is somewhat different from that of traditional books on representation theory. First, bearing in mind the needs of physicists, the author has tried to make the exposition as elementary as possible. The need for an elementary exposition has influenced the distribution of the material. The book is divided into three largely independent parts, arranged in order of increasing difficulty. Besides compact Lie groups, groups with other topological structure (“similar” to compact groups in some sense) are considered. Prominent among these are reductive complex Lie groups (including semisimple groups), obtained from compact Lie groups by analytic continuation, and also their real forms (reductive real Lie groups). The theory of finite-dimensional representation for these classes of groups is developed, striving whenever possible to emphasize the “compact origin” of these representations, i.e., their analytic relationship to representations of compact Lie groups. In conclusion, the author briefly presents some aspects of infinite-dimensional representations of semisimple complex Lie algebras and Lie groups.

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