Nilpotent Orbits, Primitive Ideals, and Characteristic Classes
Author: Walter Borho
Publisher:
Published: 1989-12-01
Total Pages: 148
ISBN-13: 9781461245599
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Nilpotent Orbits, Primitive Ideals, and Characteristic Classes
Nilpotent Orbits, Primitive Ideals, and Characteristic Classes
Author: Walter Borho
Publisher: Springer Science & Business Media
Published: 2012-12-06
Total Pages: 141
ISBN-13: 1461245583
DOWNLOAD EBOOK →1. The Subject Matter. Consider a complex semisimple Lie group G with Lie algebra g and Weyl group W. In this book, we present a geometric perspective on the following circle of ideas: polynomials The "vertices" of this graph are some of the most important objects in representation theory. Each has a theory in its own right, and each has had its own independent historical development. - A nilpotent orbit is an orbit of the adjoint action of G on g which contains the zero element of g in its closure. (For the special linear group 2 G = SL(n,C), whose Lie algebra 9 is all n x n matrices with trace zero, an adjoint orbit consists of all matrices with a given Jordan canonical form; such an orbit is nilpotent if the Jordan form has only zeros on the diagonal. In this case, the nilpotent orbits are classified by partitions of n, given by the sizes of the Jordan blocks.) The closures of the nilpotent orbits are singular in general, and understanding their singularities is an important problem. - The classification of irreducible Weyl group representations is quite old.
Nilpotent Orbits, Primitive Ideals, and Characteristic Classes
Author: Walter Borho
Publisher:
Published: 1989
Total Pages: 131
ISBN-13: 9783764334734
DOWNLOAD EBOOK →Algebraic Groups. Utrecht 1986
Author: Arjeh M. Cohen
Publisher: Springer
Published: 2006-11-15
Total Pages: 291
ISBN-13: 3540478345
DOWNLOAD EBOOK →From 1-4 April 1986 a Symposium on Algebraic Groups was held at the University of Utrecht, The Netherlands, in celebration of the 350th birthday of the University and the 60th of T.A. Springer. Recognized leaders in the field of algebraic groups and related areas gave lectures which covered wide and central areas of mathematics. Though the fourteen papers in this volume are mostly original research contributions, some survey articles are included. Centering on the Symposium subject, such diverse topics are covered as Discrete Subgroups of Lie Groups, Invariant Theory, D-modules, Lie Algebras, Special Functions, Group Actions on Varieties.
Nilpotent Orbits In Semisimple Lie Algebra
Author: William.M. McGovern
Publisher: Routledge
Published: 2017-10-19
Total Pages: 166
ISBN-13: 1351428683
DOWNLOAD EBOOK →Through the 1990s, a circle of ideas emerged relating three very different kinds of objects associated to a complex semisimple Lie algebra: nilpotent orbits, representations of a Weyl group, and primitive ideals in an enveloping algebra. The principal aim of this book is to collect together the important results concerning the classification and properties of nilpotent orbits, beginning from the common ground of basic structure theory. The techniques used are elementary and in the toolkit of any graduate student interested in the harmonic analysis of representation theory of Lie groups. The book develops the Dynkin-Konstant and Bala-Carter classifications of complex nilpotent orbits, derives the Lusztig-Spaltenstein theory of induction of nilpotent orbits, discusses basic topological questions, and classifies real nilpotent orbits. The classical algebras are emphasized throughout; here the theory can be simplified by using the combinatorics of partitions and tableaux. The authors conclude with a survey of advanced topics related to the above circle of ideas. This book is the product of a two-quarter course taught at the University of Washington.
Quantum Groups and Their Primitive Ideals
Author: Anthony Joseph
Publisher: Springer Science & Business Media
Published: 2012-12-06
Total Pages: 394
ISBN-13: 3642784003
DOWNLOAD EBOOK →by a more general quadratic algebra (possibly obtained by deformation) and then to derive Rq [G] by requiring it to possess the latter as a comodule. A third principle is to focus attention on the tensor structure of the cat egory of (!; modules. This means of course just defining an algebra structure on Rq[G]; but this is to be done in a very specific manner. Concretely the category is required to be braided and this forces (9.4.2) the existence of an "R-matrix" satisfying in particular the quantum Yang-Baxter equation and from which the algebra structure of Rq[G] can be written down (9.4.5). Finally there was a search for a perfectly self-dual model for Rq[G] which would then be isomorphic to Uq(g). Apparently this failed; but V. G. Drinfeld found that it could be essentially made to work for the "Borel part" of Uq(g) denoted U (b) and further found a general construction (the Drinfeld double) q mirroring a Lie bialgebra. This gives Uq(g) up to passage to a quotient. One of the most remarkable aspects of the above superficially different ap proaches is their extraordinary intercoherence. In particular they essentially all lead for G semisimple to the same and hence "canonical", objects Rq[G] and Uq(g), though this epithet may as yet be premature.
Algebraic Quotients. Torus Actions and Cohomology. The Adjoint Representation and the Adjoint Action
Author: A. Bialynicki-Birula
Publisher: Springer Science & Business Media
Published: 2013-03-09
Total Pages: 248
ISBN-13: 3662050714
DOWNLOAD EBOOK →This is the second volume of the new subseries "Invariant Theory and Algebraic Transformation Groups". The aim of the survey by A. Bialynicki-Birula is to present the main trends and achievements of research in the theory of quotients by actions of algebraic groups. This theory contains geometric invariant theory with various applications to problems of moduli theory. The contribution by J. Carrell treats the subject of torus actions on algebraic varieties, giving a detailed exposition of many of the cohomological results one obtains from having a torus action with fixed points. Many examples, such as toric varieties and flag varieties, are discussed in detail. W.M. McGovern studies the actions of a semisimple Lie or algebraic group on its Lie algebra via the adjoint action and on itself via conjugation. His contribution focuses primarily on nilpotent orbits that have found the widest application to representation theory in the last thirty-five years.
Conjugacy Classes in Semisimple Algebraic Groups
Author: James E. Humphreys
Publisher: American Mathematical Soc.
Published: 1995
Total Pages: 218
ISBN-13: 0821852760
DOWNLOAD EBOOK →Provides a useful exposition of results on the structure of semisimple algebraic groups over an arbitrary algebraically closed field. After the fundamental work of Borel and Chevalley in the 1950s and 1960s, further results were obtained over the next thirty years on conjugacy classes and centralizers of elements of such groups.
Lie Theory
Author: Jean-Philippe Anker
Publisher: Springer Science & Business Media
Published: 2012-12-06
Total Pages: 341
ISBN-13: 0817681922
DOWNLOAD EBOOK →* 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.
