Introduction to Lie Groups and Lie Algebra, 51 (Pure & Applied Mathematics) by Arthur A. Sagle (PDF)

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Introduction to Lie Groups and Lie Algebra, 51

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⭐This book assumes no previous knowledge of manifolds. After a short review of differential calculus in R^n, manifolds and topological groups are quickly presented. The next four chapters (4-5-6-7) deal with the Lie algebra g of left invariant vector fields on a Lie group G, the exponential map, the adjoint representation, many examples, a proof of the Baker-Campbell-Hausdorff formula (though it is not the most informative: the best is Dynkin’s proof, see Naimark-Stern’s

⭐). Chapter eight offers a quick exposition of the properties of the universal covering group of a Lie group G, ending with the proof that the group of automorphisms of a Lie group is a Lie group too. After an algebraic chapter nine, dealing with tensor products, and the complexification of a Lie algebra, we meet a couple of very good chapters on solvable and nilpotent Lie groups and their algebras, proving Engel and Lie theorems, both for groups and algebras. In chapter 12 there is a quite good exposition of semisimple Lie algebras (SSLA for short), dealing with (1) the Killing form and Cartan’s criteria for SSLA and solvable Lie algebras (shorter proofs of them in Erdmann-Wildon’s

⭐) (2) Weyl’s complete reducibility for representations (aka modules) of SSLA (3) E. Levi’s decomposition of an arbitrary Lie algebra as a semidirect product of its maximal solvable ideal and a SSLA (the proof is not complete) ; (4) Cartan subalgebras in a complex SSLA g, roots and the associated decomposition of g as a direct sum of root spaces, (5) simple roots and the Weyl group (6) the classification of complex SSLA via Dynkin diagrams (well: it is enunciated for split SSLA, a secondary and unimportant improvement). Chapter 13 deals with some elementary facts about real and complex SSLA and their representations. Little is said on complex semisimple groups or even on compact groups; but let’s remember that this book is an introduction. It’s a pity that this work has not reached a second edition; nevertheless, as it is, this book is a very clear and rigourous first encounter with Lie theory. Later, Knapp’s

⭐or Bump’s

⭐can provide further topics and bibliograpy. To continue with Lie theory, a classical and imposing reference, is Helgason’s

⭐, whose chapters 2, 3 and 4 compare well with the core of Sagle-Walde’s book, adding beautiful contrasts. But different directions of study in Lie theory exist, for example, (1) algebraic groups, see Chevalley’s foundational work

⭐, (2) simple finite groups, see Carter’s

⭐.

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