Foundations of Lie Theory and Lie Transformation Groups

The book by Gorbatsevich, Onishchik and Vinberg is the first in a series of volumes devoted to the theory of Lie groups and Lie algebras. The first part of the book deals with the foundations of the theory based on the classical global approach of Chevalley followed by an exposition of the alternative approach via the universal algebra and the Campbell-Hausdorff formula. It also contains a survey of certain generalizations of Lie groups. The second more advanced part treats the topic of Lie transformation groups covering e.g. properties of orbits and stabilizers, homogeneous fibre bundles, Frobenius duality, groups of automorphisms of geometric structures, Lie algebras of vector fields and the existence of slices. The work of the last decades including the most recent research results is covered.
The book contains numerous examples and describes connections with topology, differential geometry, analysis and applications. It will be of great interest to graduate students and researchers in mathematics and theoretical physics. From the reviews: "..., the book must be of great help for a researcher who already has some idea of Lie theory, wants to employ it in his everyday research and/or teaching, and needs a source for customary reference on the subject. From my viewpoint, the volume is perfectly fit to serve as such a source, ... On the whole, it is quite a pleasure, after making yourself comfortable in that favourite office armchair of yours, just to keep the volume gently in your hands and browse it slowly and thoughtfully; and after all, what more on Earth can one expect of any book?" The New Zealand Mathematical Society Newsletter "... Both parts are very nicely written and can be strongly recommended." European Mathematical Society I.Foundations of Lie Theory.- 1. Basic Notions.- 2. The Relation Between Lie Groups and Lie Algebras.- 3. The Universal Enveloping Algebra.- 4. Generalizations of Lie Groups.- II. Lie Transformation Groups.- 1. Lie Group Actions on Manifolds.- 2. Transitive Actions.- 3. Actions of Compact Lie Groups.- 4. Homogeneous Spaces of Nilpotent and Solvable Groups.- 5. Compact Homogeneous Spaces.- 6. Actions of Lie Groups on Low-dimensional Manifolds.