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Algebraic Quotients. Torus Actions and Cohomology. The Adjoint Representation and the Adjoint Action

  • Book
  • © 2002

Overview

Authors:
  1. Andrzej Białynicki-Birula

    1. Department of Mathematics and Computer Science, Warsaw University, Warsaw, Poland

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  2. James B. Carrell

    1. Department of Mathematics, The University of British Columbia, Vancouver, Canada

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  3. William M. McGovern

    1. Department of Mathematics, University of Washington, Seattle, USA

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  • Surveys on the current state of invariant theory
  • Includes supplementary material: sn.pub/extras

Part of the book series: Encyclopaedia of Mathematical Sciences (EMS, volume 131)

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Table of contents (3 chapters)

  1. Front Matter

    Pages i-v
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  2. Quotients by Actions of Groups

    • Andrzej Białynicki-Birula
    Pages 1-82

  3. Torus Actions and Cohomology

    • James B. Carrell
    Pages 83-158

  4. The Adjoint Representation and the Adjoint Action

    • William M. McGovern
    Pages 159-238

  5. Back Matter

    Pages 239-242
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Keywords

About this book

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.

Reviews

"This volume of the Encyclopaedia of Mathematical Sciences contains three contributions on actions of algebraic groups. The first one, by A. Bialynicki-Birula, is concerned with the general concept of a quotient, while the other two, by J.B.Carrell and W.M. McGovern, are on more specific topics. [...]

These three articles are all of value, but have somewhat different nature. Bialynicki-Birula's is a survey of a very wide area of research and requires much prior knowledge on the part of the reader. It is certainly not some something one could suggest as reading for a starting graduate student, but it is a useful reference for those who already have some knowledge and want to know about some aspect of the theory of quotients. The other two articles are much more accessible, especially that of McGovern, but could also be used as a source of general reference in the (more limited) areas they study."

P.Newstead, Liverpool, Jahresberichte der DMV, Vol. 107, Issue 4 (2005)

Authors and Affiliations

  • Department of Mathematics and Computer Science, Warsaw University, Warsaw, Poland

    Andrzej Białynicki-Birula

  • Department of Mathematics, The University of British Columbia, Vancouver, Canada

    James B. Carrell

  • Department of Mathematics, University of Washington, Seattle, USA

    William M. McGovern

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