Renner Linear Algebraic Monoids

1. Auflage 2005
ISBN: 978-3-540-27556-5
Verlag: Springer-Verlag
Format: PDF
Kopierschutz: Adobe DRM (»Systemvoraussetzungen)

E-Book, Englisch, 246 Seiten

This solid volume discusses all the key topics in detail, including classification, orbit structure, representations, universal constructions, and abstract analogues. Open problems are discussed as they arise and many useful exercises are included.

Autoren/Hrsg.

Renner, Lex

Inhaltsverzeichnis

1;Preface;6
2;Contents;8
3;1 Introduction;12
4;2 Background;16
4.1;2.1 Algebraic Geometry;16
4.2;2.2 Algebraic Groups;23
4.3;2.3 Semigroups;37
4.4;2.4 Exercises;42
5;3 Algebraic Monoids;43
5.1;3.1 Linear Algebraic Monoids;43
5.2;3.2 Normal Monoids;47
5.3;3.3 D-monoids;48
5.4;3.4 Solvable Monoids;50
5.5;3.5 Excercises;50
6;4 Regularity Conditions;54
6.1;4.1 Reductive Monoids;54
6.2;4.2 Semigroup Structure of Reductive Monoids;56
6.3;4.3 Solvable Regular Monoids;57
6.4;4.4 Regular Algebraic Monoids;59
6.5;4.5 Regularity in Codimension One;61
6.6;4.6 Exercises;63
7;5 Classification of Reductive Monoids;65
7.1;5.1 The Extension Principle;65
7.2;5.2 Vinberg’s Approach;69
7.3;5.3 Algebraic Monoids as Spherical Varieties;72
8;6 Universal Constructions;80
8.1;6.1 Quotients;80
8.2;6.2 Class Groups of Reductive Monoids;82
8.3;6.3 Flat Monoids;85
8.4;6.4 Multilined Closure;93
8.5;6.5 Normalization and Representations;96
8.6;6.6 Exercises;97
9;7 Orbit Structure of Reductive Monoids;98
9.1;7.1 The System of Idempotents and the Type Map;98
9.2;7.2 The Cross Section Lattice and the Weyl Chamber;100
9.3;7.3 J-irreducible Monoids;103
9.4;7.4 Explicit Calculations of the Type Map;106
9.5;7.5 2-reducible Reductive Monoids;114
9.6;7.6 Type Maps in General;129
9.7;7.7 Exercises;131
10;8 The Analogue of the Bruhat Decomposition;133
10.1;8.1 The Renner Monoid R;133
10.2;8.2 The Analogue of the Tits System;135
10.3;8.3 Row Reduced Echelon Form;138
10.4;8.4 The Length Function on R;141
10.5;8.5 Order-Preserving Elements of R;143
10.6;8.6 The Adherence Order on R;146
10.7;8.7 The j-order, R+ and Pennell’s Theorem;149
10.8;8.8 The Adherence Order on Mn(K);153
10.9;8.9 Exercises;156
11;9 Representations and Blocks of Algebraic Monoids;158
11.1;9.1 Conjugacy Classes and Adjoint Quotient;158
11.2;9.2 Rep(M) according to Doty;161
11.3;9.3 The Blocks of Mn(K) when char(K) = p > 0;164
11.4;9.4 The Blocks of Solvable Algebraic Monoids;165
12;10 Monoids of Lie Type;172
12.1;10.1 Finite Groups of Lie Type;172
12.2;10.2 Endomorphisms of Linear Algebraic Monoids;173
12.3;10.3 A Detailed Example;174
12.4;10.4 Abstract Monoids of Lie Type;177
12.5;10.5 Modular Representations of Finite Reductive Monoids;180
12.6;10.6 Exercises;189
13;11 Cellular Decomposition of Algebraic Monoids;191
13.1;11.1 Monoid Cells;192
13.2;11.2 Exercises;196
14;12 Conjugacy Classes;198
14.1;12.1 The Basic Conjugacy Theorem;198
14.2;12.2 Some Refinements;200
14.3;12.3 Putcha’s Decomposition and the Nilpotent Variety;202
15;13 The Centralizer of a Semisimple Element;208
15.1;13.1 Introduction;208
15.2;13.2 Main Results;209
15.3;13.3 The Structure of Rs and Ms;211
15.4;13.4 Examples;212
16;14 Combinatorics Related to Algebraic Monoids;216
16.1;14.1 The Adherence Order on WeW;216
16.2;14.2 Shellability and Stanley-Reisner Rings;220
16.3;14.3 Distribution of Products in Finite Monoids;220
16.4;14.4 Exercises;229
17;15 Survey of Related Developments;230
17.1;15.1 Complex Representation of Finite Reductive Monoids;230
17.2;15.2 Finite Semigroups and Highest Weight Categories;231
17.3;15.3 Singularities of G-embeddings;233
17.4;15.4 Cohomology of G-embeddings;234
17.5;15.5 Horospherical Varieties;235
17.6;15.6 Monoids associated with Kac-Moody Groups;235
18;References;237
19;Index;244