This is the second quarter of the year long sequence Lie Groups--Math251 ABC. The Math251A was also cross-listed as Math250A, in which the theory of smooth manifolds, vector fields, Lie derivatives, differential forms as well as the exterior derivatives and integrations of them, the Frobenius theorem concerning the integrability of distributions, connection and curvature of Riemannian metrics, etc., should have been covered. The prerequisite can also be satisfied with Chapters 1, 2, and 4 of the book by Warner Foundations of differential manifolds and Lie groups, (Scott, Foresman, 1971; Springer, 1983). Another excellent modern source for these materials is From calculus to cohomology by Madsen and Tornehave (Cambridge Univ. Press, 1997). A much shorter version for the basic materials can be obtained from two lectures of my 2016 year long course on Math250. Solid background on linear algebra, such as the polar decomposition, Jordan forms, is assumed. One can get it by reading a book such as Gelfand's Lectures on linear algebra or Shilov's Linear algebra. Both linear algebra books have cheap Dover editions.
Lie groups was introduced by Sophus Lie (1842-1899) in 19th century in an attempt to extend the Galois (1811-1832) theory for solving polynomial equations to a theory solving the differential equations. It was further developed by Killing (1847-1923), E. Cartan (1869-1851, Thesis 1894), H. Weyl (1885-1955) and many other prominent mathematicians into a coherent and sophisticated branch of mathematics which has found its applications in many areas of modern mathematics and physics such as algebraic topology, differential geometry, calculus variation and PDEs, invariant theory, bifurcation theory, control theory, classical and quantum mechanics, relativity, etc. Cartan's study led him to the classification of symmetric spaces, namely the spaces with constant curvature, which is one of the crowning achievements of modern geometry.
The concept of Lie algebra, which arises as the tangent space of a Lie group, was initially coined by H. Weyl. The abstraction and its independent study as a subject not only are important in understanding Lie groups, but also turn out to be very fruitful in the study of algebraic groups, quantum groups, combinatorics and geometry.
Any mathematical proof in serious mathematical textbooks as put by A. Y. Khinchin ` will undoubtedly seem very complicated to you. But it will take you only two to three week's work with pencil and paper to understand and digest it completely. It is by conquering difficulties of just this sort, that the mathematicians/or mathematical students grow and develop.' Even though none of the theorems involved in this course is a difficult one requiring the labor beyond several hours to digest, the same principle on the effort part applies. No good mathematics can be spoon-fed fast.
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Required Text:
IT = Lie groups I, II, by Ise and Takeuchi, AMS, Translation Series, 1991;
Ch = Lie groups, by Chevalley, Princeton, 1946;
Ziller = Lie groups, Representation Theory and Symmetric Spaces, Ziller's Notes.
Recommended other Texts:
Pon= Topological groups, by Pontryagin, 2nd, Gordon and Breach, 1966, Ch 7, 10, 11;
Hsi= Lectures on (compact) Lie groups, by Hsiang, World Scientific, 1998;
Complex semisimple Lie algebras, by Serre, French version, Benjamin, 1966; English version, Springer, 1987;
Hum= Introduction to Lie algebras and its representations, by Humphreys, Springer, 1972;
Harmonic analysis on homogenous spaces,, by Wallach, 2nd Edition, Dover, 2018, Ch 1-4.
Lie groups ,,lecture notes by Donaldson
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