Overview

Homogeneous spaces of linear algebraic groups lie at the crossroads of algebraic geometry, theory of algebraic groups, classical projective and enumerative geometry, harmonic analysis, and representation theory. By standard reasons of algebraic geometry, in order to solve various problems on a homogeneous space, it is natural and helpful to compactify it while keeping track of the group action, i.e., to consider equivariant completions or, more generally, open embeddings of a given homogeneous space. Such equivariant embeddings are the subject of this book. We focus on the classification of equivariant embeddings in terms of certain data of ""combinatorial"" nature (the Luna-Vust theory) and description of various geometric and representation-theoretic properties of these varieties based on these data. The class of spherical varieties, intensively studied during the last three decades, is of special interest in the scope of this book. Spherical varieties include many classical examples, such as Grassmannians, flag varieties, and varieties of quadrics, as well as well-known toric varieties. We have attempted to cover most of the important issues, including the recent substantial progress obtained in and around the theory of spherical varieties.

Full Product Details

Author:   D.A. Timashev
Publisher:   Springer-Verlag Berlin and Heidelberg GmbH & Co. KG
Imprint:   Springer-Verlag Berlin and Heidelberg GmbH & Co. K
Edition:   2011 ed.
Volume:   138
Dimensions:   Width: 15.50cm , Height: 2.30cm , Length: 23.50cm
Weight:   0.604kg
ISBN:  

9783642183980

ISBN 10:   3642183980
Pages:   254
Publication Date:   07 April 2011
Audience:   Professional and scholarly ,  Professional & Vocational
Format:   Hardback
Publisher's Status:   Active
Availability:   Awaiting stock  
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Table of Contents

Introduction.- 1 Algebraic Homogeneous Spaces.- 2 Complexity and Rank.- 3 General Theory of Embeddings.- 4 Invariant Valuations.- 5 Spherical Varieties.- Appendices.- Bibliography.- Indices

Reviews

From the reviews: This book is a survey of the study of equivariant embeddings of homogeneous spaces for connected reductive algebraic groups. ... The book is well written and puts together many results in a complete and concise manner. (Lucy Moser-Jauslin, Mathematical Reviews, Issue 2012 e) The book under review is concerned with the study of equivariant embeddings of homogeneous spaces ... . This monograph provides a compact survey of these issues, including many results of Brion, Knop, Luna, and Vust, among others, as well as the author's own significant contributions to the field. ... The author admirably fills a gap in the current literature on algebraic groups with this book. It should make a welcome addition to any mathematician's library. (Nathan Ilten, Zentralblatt MATH, Vol. 1237, 2012)

From the reviews: This book is a survey of the study of equivariant embeddings of homogeneous spaces for connected reductive algebraic groups. ... The book is well written and puts together many results in a complete and concise manner. (Lucy Moser-Jauslin, Mathematical Reviews, Issue 2012 e)

From the reviews: The book by D. A. Timashev is a welcome survey of old and new results on actions of algebraic groups on algebraic varieties. ... The book contains an extensive list of references, both to original research articles as well as to books and monographs. It is surely a welcome guide for nonexperts who want to enter the subject and a good reference for the specialist. (Gestur Olafsson, Bulletin of the American Mathematical Society, Vol. 51 (2), April, 2014) This book is a survey of the study of equivariant embeddings of homogeneous spaces for connected reductive algebraic groups. ... The book is well written and puts together many results in a complete and concise manner. (Lucy Moser-Jauslin, Mathematical Reviews, Issue 2012 e) The book under review is concerned with the study of equivariant embeddings of homogeneous spaces ... . This monograph provides a compact survey of these issues, including many results of Brion, Knop, Luna, and Vust, among others, as well as the author's own significant contributions to the field. ... The author admirably fills a gap in the current literature on algebraic groups with this book. It should make a welcome addition to any mathematician's library. (Nathan Ilten, Zentralblatt MATH, Vol. 1237, 2012)

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