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Representations of Algebraic Groups
Gives an introduction to the general theory of representations of algebraic group schemes. This title deals with representation theory of reductive algebraic groups and includes topics such as the description of simple modules, vanishing theorems, Borel-Bott-Weil theorem and Weyl's character formula, and Schubert schemes and lne bundles on them.
Lectures on Quantum Groups
Starting with the quantum analog of sl2, the author carefully leads the reader through all the details necessary for full understanding of the subject, particularly emphasizing similarities and differences with the classical theory. The final chapters of the book describe the Kashiwara-Lusztig theory of so-called crystal (or canonical) bases in representations of complex semisimple Lie algebra.
Lie Theory
* First of three independent, self-contained volumes under the general title, "Lie Theory," featuring original results and survey work from renowned mathematicians. * Contains J. C. Jantzen's "Nilpotent Orbits in Representation Theory," and K.-H. Neeb's "Infinite Dimensional Groups and their Representations." * Comprehensive treatments of the relevant geometry of orbits in Lie algebras, or their duals, and the correspondence to representations. * Should benefit graduate students and researchers in mathematics and mathematical physics.
Representations of Reductive Groups
This volume provides a very accessible introduction to the representation theory of reductive algebraic groups.
Introduction to Quantum Groups
The quantum groups discussed in this book are the quantized enveloping algebras introduced by Drinfeld and Jimbo in 1985, or variations thereof. The theory of quantum groups has led to a new, extremely rigid structure, in which the objects of the theory are provided with canonical basis with rather remarkable properties. This book will be of interest to mathematicians working in the representation theory of Lie groups and Lie algebras, knot theorists and to theoretical physicists and graduate students. Since large parts of the book are independent of the theory of perverse sheaves, the book could also be used as a text book.
Groups and Analysis
Many areas of mathematics were deeply influenced or even founded by Hermann Weyl, including geometric foundations of manifolds and physics, topological groups, Lie groups and representation theory, harmonic analysis and analytic number theory as well as foundations of mathematics. In this volume, leading experts present his lasting influence on current mathematics, often connecting Weyl's theorems with cutting edge research in dynamical systems, invariant theory, and partial differential equations. In a broad and accessible presentation, survey chapters describe the historical development of each area alongside up-to-the-minute results, focussing on the mathematical roots evident within Weyl's work.
Representations of Semisimple Lie Algebras in the BGG Category O
This is the first textbook treatment of work leading to the landmark 1979 Kazhdan–Lusztig Conjecture on characters of simple highest weight modules for a semisimple Lie algebra g g over C C. The setting is the module category O O introduced by Bernstein–Gelfand–Gelfand, which includes all highest weight modules for g g such as Verma modules and finite dimensional simple modules. Analogues of this category have become influential in many areas of representation theory. Part I can be used as a text for independent study or for a mid-level one semester graduate course; it includes exercises and examples. The main prerequisite is familiarity with the structure theory of g g. Basic techniqu...
Representation Theories and Algebraic Geometry
The 12 lectures presented in Representation Theories and Algebraic Geometry focus on the very rich and powerful interplay between algebraic geometry and the representation theories of various modern mathematical structures, such as reductive groups, quantum groups, Hecke algebras, restricted Lie algebras, and their companions. This interplay has been extensively exploited during recent years, resulting in great progress in these representation theories. Conversely, a great stimulus has been given to the development of such geometric theories as D-modules, perverse sheafs and equivariant intersection cohomology. The range of topics covered is wide, from equivariant Chow groups, decomposition classes and Schubert varieties, multiplicity free actions, convolution algebras, standard monomial theory, and canonical bases, to annihilators of quantum Verma modules, modular representation theory of Lie algebras and combinatorics of representation categories of Harish-Chandra modules.
Algebraic Groups and Their Representations
This volume contains articles by 20 leading workers in the field of algebraic groups and related finite groups. Articles on representation theory are written by Andersen on tilting modules, Carter on canonical bases, Cline, Parshall and Scott on endomorphism algebras, James and Kleshchev on the symmetric group, Littelmann on the path model, Lusztig on homology bases, McNinch on semisimplicity in prime characteristic, Robinson on block theory, Scott on Lusztig's character formula, and Tanisaki on highest weight modules. Articles on subgroup structure are written by Seitz and Brundan on double cosets, Liebeck on exceptional groups, Saxl on subgroups containing special elements, and Guralnick on applications of subgroup structure. Steinberg gives a new, short proof of the isomorphism and isogeny theorems for reductive groups. Aschbacher discusses the classification of quasithin groups and Borovik the classification of groups of finite Morley rank. Audience: The book contains accounts of many recent advances and will interest research workers and students in the theory of algebraic groups and related areas of mathematics.
