Kac-Moody Lie algebras 9 were introduced in the mid-1960s independently by V. The theory has undergone tremendous developments in various directions and connections with diverse areas abound, including mathematical physics, so much so that this theory has become a stan­ dard tool in mathematics.

I. Kac-Moody Algebras: Basic Theory.- 1. Definition of Kac-Moody Algebras.- 2. Root Space Decomposition.- 3. Weyl Groups Associated to Kac-Moody Algebras.- 4. Dominant Chamber and Tits Cone.- 5. Invariant Bilinear Form and the Casimir Operator.- II. Representation Theory of Kac-Moody Algebras.- 1. Category $$\mathcal{O}$$.- 2. Weyl-Kac Character Formula.- 3. Shapovalov Bilinear Form.- III. Lie Algebra Homology and Cohomology.- 1. Basic Definitions and Elementary Properties.- 2. Lie Algebra Homology of n-: Results of Kostant-Garland-Lepowsky.- 3. Decomposition of the Category $$\mathcal{O}$$ and some Ext Vanishing Results.- 4. Laplacian Calculation.- IV. An Introduction to ind-Varieties and pro-Groups.- 1. Ind-Varieties: Basic Definitions.- 2. Ind-Groups and their Lie Algebras.- 3. Smoothness of ind-Varieties.- 4. An Introduction to pro-Groups and pro-Lie Algebras.- V. Tits Systems: Basic Theory.- 1. An Introduction to Tits Systems.- 2. Refined Tits Systems.- VI. Kac-Moody Groups: Basic Theory.- 1. Definition of Kac-Moody Groups and Parabolic Subgroups.- 2. Representations of Kac-Moody Groups.- VII. Generalized Flag Varieties of Kac-Moody Groups.- 1. Generalized Flag Varieties: Ind-Variety Structure.- 2. Line Bundles on $${\mathcal{X}^Y}$$.- 3. Study of the Group $${\mathcal{U}^ - }$$.- 4. Study of the Group $${\mathcal{G}^{\min }}$$ Defined by Kac-Peterson.- VIII. Demazure and Weyl-Kac Character Formulas.- 1. Cohomology of Certain Line Bundles on $${Z_\mathfrak{w}}$$.- 2. Normality of Schubert Varieties and the Demazure Character Formula.- 3. Extension of the Weyl-Kac Character Formula and the Borel-Weil-Bott Theorem.- IX. BGG and Kempf Resolutions.- 1. BGG Resolution: Algebraic Proof in the Symmetrizable Case.- 2. A Combinatorial Description of the BGG Resolution.- 3.Kempf Resolution.- X. Defining Equations of $$\mathcal{G}/\mathcal{P}$$ and Conjugacy Theorems.- 1. Quadratic Generation of Defining Ideals of $$\mathcal{G}/\mathcal{P}$$ in Projective Embeddings.- 2. Conjugacy Theorems for Lie Algebras.- 3. Conjugacy Theorems for Groups.- XI. Topology of Kac-Moody Groups and Their Flag Varieties.- 1. The Nil-Hecke Ring.- 2. Determination of $$\bar R$$.- 3. T-equivariant Cohomology of $$\mathcal{G}/\mathcal{P}$$.- 4. Positivity of the Cup Product in the Cohomology of Flag Varieties.- 5. Degeneracy of the Leray-Serre Spectral Sequence for the Fibration $${\mathcal{G}^{\min }} \to {\mathcal{G}^{\min }}/T$$.- XII. Smoothness and Rational Smoothness of Schubert Varieties.- 1. Singular Locus of Schubert Varieties.- 2. Rational Smoothness of Schubert Varieties.- XIII. An Introduction to Affine Kac-Moody Lie Algebras and Groups.- 1. Affine Kac-Moody Lie Algebras.- 2. Affine Kac-Moody Groups.- Appendix A. Results from Algebraic Geometry.- Appendix B. Local Cohomology.- Appendix C. Results from Topology.- Appendix D. Relative Homological Algebra.- Appendix E. An Introduction to Spectral Sequences.- Index of Notation.

"Most of these topics appear here for the first time in book form. Many of them are interesting even in the classical case of semi-simple algebraic groups. Some appendices recall useful results from other areas, so the work may be considered self-contained, although some familiarity with semi-simple Lie algebras or algebraic groups is helpful. It is clear that this book is a valuable reference for all those interested in flag varieties and representation theory in the semi-simple or Kac-Moody case." —MATHEMATICAL REVIEWS "A lot of different topics are treated in this monumental work. . . . many of the topics of the book will be useful for those only interested in the finite-dimensional case. The book is self contained, but is on the level of advanced graduate students. . . . For the motivated reader who is willing to spend considerable time on the material, the book can be a gold mine. " —ZENTRALBLATT MATH