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

Autoren/Hrsg.

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