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**Lie groups. An introduction through linear groups.**
*(English)*
Zbl 0989.22001

Oxford Graduate Texts in Mathematics. 5. Oxford: Oxford University Press. x, 265 p. (2002).

This is an excellent and original introductory course on Lie groups. The author starts with an exposition of Lie theory in the context of groups of real or complex matrices: one-parameter groups and the exponential map, the correspondence between linear groups and linear Lie algebras, classical groups. The following topics exposed with details should be mentioned: the Dynkin formula for the exponential map proved by following Duistermaat-Kolk, the 1-1 correspondence between connected linear groups and linear Lie algebras, roots, weights and diagrams of classical groups including their fundamental groups. In this part of the book, only the understanding of linear algebra and multivariable calculus is required. As the second step, analytic manifolds and Lie groups are introduced and their analytic actions, in particular, homogeneous spaces, are studied. Integration on manifolds is defined. The last part presents an introduction to representation theory, including the Weyl character formula and the Borel-Weil theorem for classical groups. Each section is supplied with numerous problems.

Reviewer: A.L.Onishchik (Yaroslavl)

### MSC:

22-02 | Research exposition (monographs, survey articles) pertaining to topological groups |

22Exx | Lie groups |