Holdings: Algebraic quotients

Algebraic quotients:

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Bibliographic Details
Main Authors:	Białynicki-Birula, Andrzej (Author), Carrell, James B. (Author), McGovern, William M. 1959- (Author)
Format:	Book
Language:	English
Published:	Berlin [u.a.] Springer 2002
Series:	Encyclopaedia of mathematical sciences 131
Subjects:	Geometria algébrica Homologie Lie, Algèbres de Lie, Groupes de Teoria geométrica de invariantes Torsion, Théorie de la (Algèbre) Homology theory Lie algebras Lie groups Quotient rings Torsion theory (Algebra) Geometrische Invariantentheorie Gruppenoperation Algebraische Gruppe
Online Access:	Inhaltsverzeichnis
Item Description:	Enth. u.a.: Torus actions and cohomology / J. B. Carrell
Physical Description:	242 S. graph. Darst.
ISBN:	3540432116

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245	1	0	\|a Algebraic quotients \|c A. Białynicki-Birula ; J. B. Carrell ; W. M. McGovern
246	1	3	\|a Nebent.: Quotients by actions of groups
264		1	\|a Berlin [u.a.] \|b Springer \|c 2002
300			\|a 242 S. \|b graph. Darst.
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490	1		\|a Encyclopaedia of mathematical sciences \|v 131
490	1		\|a Encyclopaedia of mathematical sciences / Invariant theory and algebraic transformation groups \|v 2
500			\|a Enth. u.a.: Torus actions and cohomology / J. B. Carrell
650		7	\|a Geometria algébrica \|2 larpcal
650		4	\|a Homologie
650		4	\|a Lie, Algèbres de
650		4	\|a Lie, Groupes de
650		7	\|a Teoria geométrica de invariantes \|2 larpcal
650		4	\|a Torsion, Théorie de la (Algèbre)
650		4	\|a Homology theory
650		4	\|a Lie algebras
650		4	\|a Lie groups
650		4	\|a Quotient rings
650		4	\|a Torsion theory (Algebra)
650	0	7	\|a Geometrische Invariantentheorie \|0 (DE-588)4156712-2 \|2 gnd \|9 rswk-swf
650	0	7	\|a Gruppenoperation \|0 (DE-588)4158467-3 \|2 gnd \|9 rswk-swf
650	0	7	\|a Algebraische Gruppe \|0 (DE-588)4001164-1 \|2 gnd \|9 rswk-swf
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689	0	1	\|a Gruppenoperation \|0 (DE-588)4158467-3 \|D s
689	0	2	\|a Geometrische Invariantentheorie \|0 (DE-588)4156712-2 \|D s
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700	1		\|a Carrell, James B. \|e Verfasser \|4 aut
700	1		\|a McGovern, William M. \|d 1959- \|e Verfasser \|0 (DE-588)124769179 \|4 aut
700	1	2	\|a Białynicki-Birula, Andrzej \|4 aut \|t Quotients by actions of groups
700	1	2	\|a Carrell, James B. \|4 aut \|t Torus actions and cohomology
810	2		\|a Invariant theory and algebraic transformation groups \|t Encyclopaedia of mathematical sciences \|v 2 \|w (DE-604)BV014336202 \|9 2
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Record in the Search Index

_version_	1804129234784878592
adam_text	CONTENTS I. QUOTIENTS BY ACTIONS OF GROUPS ANDRZEJ BIAYNICKI-BIRULA 1 II. TORUS ACTIONS AND COHOMOLOGY JAMES B. CARRELL 83 III. THE ADJOINT REPRESENTATION AND THE ADJOINT ACTION WILLIAM M. MCGOVERN 159 SUBJECT INDEX 239 I. QUOTIENTS BY ACTIONS OF GROUPS ANDRZEJ BIAYNICKI-BIRULA CONTENTS CHAPTER 1 INTRODUCTION . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 CHAPTER 2 TERMINOLOGY AND NOTATION . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 §2.1. SPACES, SCHEMES, VARIETIES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 §2.2. PREEQUIVALENCE RELATIONS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 §2.3. GROUPOIDS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 §2.4. GROUP ACTIONS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 CHAPTER 3 BASIC DEFINITIONS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 CHAPTER 4 STANDARD EXAMPLES AND APPLICATIONS . . . . . . . . . . . . . . . . . . . . . . . . 12 §4.1. PRINCIPAL BUNDLES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 §4.2. PROJECTIVE SPACES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 §4.3. QUOTIENTS BY FINITE GROUP ACTIONS. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 §4.4. COSET VARIETIES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14 §4.5. QUOTIENTS OF AFFINE SPACES BY LINEAR ACTIONS OF GROUPS . . . . . . . . . . 16 §4.6. INDUCED ACTIONS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16 CHAPTER 5 THE AFFINE CASE . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17 CHAPTER 6 MUMFORDS G.I.T. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18 §6.1. GENERAL CASE . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19 §6.2. CASE OF A PROJECTIVE SPACE . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20 2 ANDRZEJ BIAYNICKI-BIRULA §6.3. CASE OF A PROJECTIVE VARIETY . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21 §6.4. HILBERTMUMFORD CRITERIUM . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21 §6.5. THE CASE OF ACTIONS OF ANY LINEAR ALGEBRAIC GROUP . . . . . . . . . . . . . 22 CHAPTER 7 GOOD QUOTIENTS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23 §7.1. GENERAL PROPERTIES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23 §7.2. G -MAXIMAL SUBSETS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25 §7.3. VARIATION OF STABILITY . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27 CHAPTER 8 PROPER ACTIONS AND QUOTIENTS BY PROPER ACTIONS . . . . . . . . . . . . . . 28 §8.1. QUOTIENTS BY CLOSED EQUIVALENCE RELATIONS . . . . . . . . . . . . . . . . . . . . . 28 §8.2. EXISTENCE OF QUOTIENTS BY PROPER GROUP ACTIONS . . . . . . . . . . . . . . . . . 29 §8.3. EXISTENCE OF QUOTIENTS OF PROPER GROUPOIDS . . . . . . . . . . . . . . . . . . . . . 30 §8.4. SESHADRI COVER . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 CHAPTER 9 STACKS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 CHAPTER 10 COMPLEX ANALYTIC CASE . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 §10.1.QUOTIENTS BY PROPER ACTIONS AND EQUIVALENCE RELATIONS . . . . . . . . . . . 37 §10.2.STEIN AND GOOD QUOTIENTS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 §10.3.MOMENT MAPS. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 CHAPTER 11 GOOD QUOTIENTS BY ACTIONS OF TORI . . . . . . . . . . . . . . . . . . . . . . . . . . 42 §11.1.QUOTIENTS BY ACTIONS OF ONE-DIMENSIONAL TORI . . . . . . . . . . . . . . . . . . 42 §11.2.QUOTIENTS BY ACTIONS OF TORI OF ARBITRARY DIMENSION . . . . . . . . . . . . . 47 §11.3.QUOTIENTS OF OPEN SUBSETS OF PROJECTIVE AND AFFINE SPACES . . . . . . . . 51 §11.4.GOOD QUOTIENTS OF TORIC VARIETIES BY ACTIONS OF TORI . . . . . . . . . . . . . 53 CHAPTER 12 HILBERTMUMFORD TYPE THEOREMS . . . . . . . . . . . . . . . . . . . . . . . . . . 54 CHAPTER 13 CHOW AND HILBERT QUOTIENTS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56 CHAPTER 14 CATEGORICAL QUOTIENTS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57 §14.1.CATEGORICAL QUOTIENTS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57 §14.2.QUOTIENT MORPHISMS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59 CHAPTER 15 SECTIONS, SLICES AND REDUCTIONS . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59 CHAPTER 16 LOCAL AND GLOBAL PROPERTIES OF QUOTIENTS. . . . . . . . . . . . . . . . . . . . . 64 II. TORUS ACTIONS AND COHOMOLOGY JAMES B. CARRELL CONTENTS CHAPTER 1 INTRODUCTION . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85 CHAPTER 2 SOME COMMENTS ON T -VARIETIES . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87 §2.1. PRELIMINARY DEFINITIONS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87 §2.2. TORUS ACTIONS ON VARIETIES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89 §2.3. TORIC VARIETIES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93 §2.4. PROJECTIVE TORUS ORBIT CLOSURES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95 CHAPTER 3 TORUS ACTIONS IN LIE THEORY . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96 §3.1. THE LIE ALGEBRA . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96 §3.2. THE ROOT SYSTEM AND THE WEYL GROUP . . . . . . . . . . . . . . . . . . . . . . . . . . 97 §3.3. NILPOTENT ORBIT CLOSURES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99 §3.4. GENERALIZED FLAG VARIETIES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102 CHAPTER 4 TORUS ACTIONS AND HOMOLOGY . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108 §4.1. THE BIAYNICKI-BIRULA DECOMPOSITION . . . . . . . . . . . . . . . . . . . . . . . . . . 109 §4.2. TORUS ACTIONS AND Z -HOMOLOGY . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112 §4.3. GOOD DECOMPOSITIONS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117 CHAPTER 5 TORUS ACTIONS AND COHOMOLOGY ALGEBRAS . . . . . . . . . . . . . . . . . . . . 119 §5.1. THE BOTT RESIDUE FORMULA . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 126 §5.2. COHOMOLOGY AND B -ACTIONS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129 84 JAMES B. CARRELL CHAPTER 6 COHOMOLOGY OF INVARIANT SUBVARIETIES . . . . . . . . . . . . . . . . . . . . . . . 133 §6.1. NON-ISOLATED FIXED POINTS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133 §6.2. COHOMOLOGY OF B -INVARIANT SUBVARIETIES . . . . . . . . . . . . . . . . . . . . . . . 139 §6.3. T -EQUIVARIANT COHOMOLOGY . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 144 §6.4. RATIONAL SMOOTHNESS AND POINCAR´ E DUALITY . . . . . . . . . . . . . . . . . . . . . . 147 §6.5. EQUIVARIANT MULTIPLICITIES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 151 REFERENCES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 154 III. THE ADJOINT REPRESENTATION AND THE ADJOINT ACTION WILLIAM M. MCGOVERN CONTENTS CHAPTER 1 PRELIMINARIES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 162 §1.1. STRUCTURE OF SEMISIMPLE LIE ALGEBRAS . . . . . . . . . . . . . . . . . . . . . . . . . . 162 §1.2. CLASSIFICATION OF SEMISIMPLE LIE ALGEBRAS . . . . . . . . . . . . . . . . . . . . . . 163 §1.3. REAL SEMISIMPLE LIE ALGEBRAS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 163 §1.4. SEMISIMPLE ALGEBRAIC GROUPS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 165 §1.5. PARABOLIC SUBALGEBRAS AND SUBGROUPS . . . . . . . . . . . . . . . . . . . . . . . . . . 166 CHAPTER 2 BASIC FACTS ABOUT ORBITS, CLASSES AND CENTRALIZERS . . . . . . . . . . . . 167 §2.1. EXAMPLE: GENERAL AND SPECIAL LINEAR GROUPS . . . . . . . . . . . . . . . . . . . 167 §2.2. EVENNESS OF DIMENSION IN GOOD CHARACTERISTIC . . . . . . . . . . . . . . . . . . 169 §2.3. ORBITS AND CENTRALIZERS OF SEMISIMPLE ELEMENTS . . . . . . . . . . . . . . . . . 170 §2.4. SEMISIMPLE ORBITS IN CLASSICAL LIE ALGEBRAS . . . . . . . . . . . . . . . . . . . . 171 §2.5. SEMISIMPLE ORBITS AND THE ADJOINT QUOTIENT . . . . . . . . . . . . . . . . . . . . 172 CHAPTER 3 NILPOTENT ORBITS AND UNIPOTENT CLASSES: THE FINITENESS THEOREM . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 174 §3.1. NILPOTENT ORBITS IN CLASSICAL LIE ALGEBRAS . . . . . . . . . . . . . . . . . . . . . . 174 §3.2. THE JACOBSONMOROZOV THEOREM . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 176 §3.3. NILPOTENT ORBITS IN REAL CLASSICAL LIE ALGEBRAS . . . . . . . . . . . . . . . . . . 178 §3.4. CONJUGACY THEOREMS AND WEIGHTED DIAGRAMS . . . . . . . . . . . . . . . . . . . 180 160 WILLIAM M. MCGOVERN §3.5. CENTRALIZERS OF NILPOTENT ELEMENTS . . . . . . . . . . . . . . . . . . . . . . . . . . . . 182 §3.6. FINITENESS THEOREMS OF RICHARDSON AND LUSZTIG . . . . . . . . . . . . . . . . . 184 §3.7. THE FINITENESS THEOREM OVER R . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 185 CHAPTER 4 THE PRINCIPAL NILPOTENT ORBIT AND REGULAR ELEMENTS . . . . . . . . . . . 187 §4.1. THE PRINCIPAL NILPOTENT ORBIT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 188 §4.2. REGULAR ELEMENTS AND THE ADJOINT QUOTIENT . . . . . . . . . . . . . . . . . . . . . 189 §4.3. RESULTS OF KOSTANT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 190 CHAPTER 5 INDUCTION OF ORBITS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 192 §5.1. INDUCED ORBITS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 192 §5.2. CRITERION FOR AN ORBIT TO BE INDUCED . . . . . . . . . . . . . . . . . . . . . . . . . . . . 194 §5.3. INDUCED ORBITS IN CLASSICAL LIE ALGEBRAS . . . . . . . . . . . . . . . . . . . . . . . 195 §5.4. APPLICATION: THE SUBREGULAR ORBIT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 196 §5.5. SECOND APPLICATION: SHEETS IN SEMISIMPLE LIE ALGEBRAS . . . . . . . . . . 197 §5.6. THIRD APPLICATION: BALACARTER CLASSIFICATION OF NILPOTENT ORBITS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 198 §5.7. APPENDIX: TABLES OF THE EXCEPTIONAL ORBITS . . . . . . . . . . . . . . . . . . . . . 200 CHAPTER 6 CLOSURES OF ORBITS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 208 §6.1. THE MINIMAL NILPOTENT ORBIT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 208 §6.2. ORBIT CLOSURES IN CLASSICAL LIE ALGEBRAS . . . . . . . . . . . . . . . . . . . . . . . . 208 §6.3. SPALTENSTEIN DUALITY . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 210 §6.4. TABLES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 212 CHAPTER 7 THE NILPOTENT VARIETY AND THE FLAG VARIETY . . . . . . . . . . . . . . . . . . . 215 §7.1. SPRINGERS DESINGULARIZATION OF N . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 216 §7.2. CONNECTEDNESS OF THE FIBERS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 217 §7.3. EQUIDIMENSIONALITY OF THE FIBERS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 217 §7.4. DIMENSIONS OF THE FIBERS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 219 §7.5. ORBITAL VARIETIES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 220 §7.6. COMPONENTS OF B X : THE SUBREGULAR CASE . . . . . . . . . . . . . . . . . . . . . . . 221 §7.7. COMPONENTS OF B X : THE CLASSICAL CASE. . . . . . . . . . . . . . . . . . . . . . . . . 222 CHAPTER 8 SPRINGERS WEYL GROUP REPRESENTATIONS . . . . . . . . . . . . . . . . . . . . . . 223 §8.1. THE SPRINGER CORRESPONDENCE. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 224 §8.2. OTHER CONSTRUCTIONS OF THE SPRINGER REPRESENTATIONS. . . . . . . . . . . . . . 225 §8.3. COMPUTATION OF THE SPRINGER REPRESENTATIONS . . . . . . . . . . . . . . . . . . . . 226 III. THE ADJOINT REPRESENTATION AND THE ADJOINT ACTION 161 CHAPTER 9 RECENT WORK . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 227 §9.1. REGULAR FUNCTIONS ON NILPOTENT ORBITS . . . . . . . . . . . . . . . . . . . . . . . . . 228 §9.2. SOMMERSS COMPUTATION OF FUNDAMENTAL GROUPS . . . . . . . . . . . . . . . . 229 §9.3. SMALL REPRESENTATIONS AND NILPOTENT ORBITS . . . . . . . . . . . . . . . . . . . . . 230 §9.4. ORBITAL VARIETIES IN CLASSICAL LIE ALGEBRAS . . . . . . . . . . . . . . . . . . . . . . 231 §9.5. NILPOTENT ORBITS IN DOUBLED LIE ALGEBRAS . . . . . . . . . . . . . . . . . . . . . . . 232 REFERENCES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 232 PREFACE THE PURPOSE OF THIS ARTICLE IS TO STUDY IN DETAIL THE ACTIONS OF A SEMISIMPLE LIE OR ALGEBRAIC GROUP ON ITS LIE ALGEBRA BY THE ADJOINT REPRESENTATION AND ON ITSELF BY THE ADJOINT ACTION. WE WILL FOCUS PRIMARILY ON ORBITS THROUGH NILPOTENT ELEMENTS IN THE LIE ALGEBRA; THESE ARE CALLED NILPOTENT ORBITS FOR SHORT. MANY DEEP RESULTS ABOUT SUCH ORBITS HAVE BEEN OBTAINED IN THE LAST THIRTY-FIVE YEARS; WE WILL COLLECT SOME OF THE MOST SIGNIFICANT OF THESE THAT HAVE FOUND WIDE APPLICATION TO REPRESENTATION THEORY. WE WILL PRIMARILY WORK IN THE SETTING OF A SEMISIMPLE LIE ALGEBRA AND ITS ADJOINT GROUP OVER AN ALGEBRAICALLY CLOSED FIELD OF CHARACTERISTIC ZERO, BUT WE WILL EXTEND MUCH OF WHAT WE DO TO SEMISIMPLE LIE ALGEBRAS OVER THE REALS OR AN ALGEBRAICALLY CLOSED FIELD OF PRIME CHARACTERISTIC, AND TO CONJUGACY CLASSES IN SEMISIMPLE ALGEBRAIC GROUPS. WE WILL GIVE DETAILED PROOFS OF MANY RESULTS, INCLUDING SOME WHICH ARE DIFFICULT TO FERRET OUT OF THE LITERATURE. OTHER RESULTS WILL BE SUMMARIZED WITH REASONABLY COMPLETE REFERENCES. THE TREATMENT IS A MORE COMPREHENSIVE VERSION OF THAT IN [CM93]; THERE IS ALSO SOME OVERLAP WITH HUMPHREYS*S BOOK [HU95]. IN THE LAST CHAPTER WE SUMMARIZE SOME OF THE MOST RECENT WORK BEING DONE IN THIS TOPIC AND INDICATE SOME DIRECTIONS OF CURRENT RESEARCH. THE READER IS EXPECTED TO BE FAMILIAR WITH THE STRUCTURE AND CLASSIFICATION OF COMPLEX SEMISIMPLE LIE ALGEBRAS, TOGETHER WITH THE BASIC DEFINITIONS AND THEOREMS TYPICALLY FOUND IN A FIRST COURSE ON THAT SUBJECT. WE WILL ALSO INVOKE THE CORRESPONDING FACTS ABOUT REAL LIE GROUPS AND ALGEBRAS AND ALGEBRAIC GROUPS FROM TIME TO TIME. FOR CONVENIENCE THIS BACKGROUND MATERIAL IS SUMMARIZED IN CHAPTER 1. THE CLASSICAL MATRIX GROUPS AND ALGEBRAS WILL SERVE AS A READY SOURCE OF EXAMPLES; WE WILL OFTEN BE ABLE TO DERIVE VERY EXPLICIT RESULTS FOR SUCH GROUPS AND ALGEBRAS USING NOTHING MORE THAN LINEAR ALGEBRA.
any_adam_object	1
author	Białynicki-Birula, Andrzej Carrell, James B. McGovern, William M. 1959- Białynicki-Birula, Andrzej Carrell, James B.
author_GND	(DE-588)124769179
author_facet	Białynicki-Birula, Andrzej Carrell, James B. McGovern, William M. 1959- Białynicki-Birula, Andrzej Carrell, James B.
author_role	aut aut aut aut aut
author_sort	Białynicki-Birula, Andrzej
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building	Verbundindex
bvnumber	BV014336247
callnumber-first	Q - Science
callnumber-label	QA251
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dewey-sort	3512.2
dewey-tens	510 - Mathematics
discipline	Mathematik
format	Book
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illustrated	Illustrated
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language	English
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physical	242 S. graph. Darst.
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series	Encyclopaedia of mathematical sciences
series2	Encyclopaedia of mathematical sciences Encyclopaedia of mathematical sciences / Invariant theory and algebraic transformation groups
spelling	Białynicki-Birula, Andrzej Verfasser aut Algebraic quotients A. Białynicki-Birula ; J. B. Carrell ; W. M. McGovern Nebent.: Quotients by actions of groups Berlin [u.a.] Springer 2002 242 S. graph. Darst. txt rdacontent n rdamedia nc rdacarrier Encyclopaedia of mathematical sciences 131 Encyclopaedia of mathematical sciences / Invariant theory and algebraic transformation groups 2 Enth. u.a.: Torus actions and cohomology / J. B. Carrell Geometria algébrica larpcal Homologie Lie, Algèbres de Lie, Groupes de Teoria geométrica de invariantes larpcal Torsion, Théorie de la (Algèbre) Homology theory Lie algebras Lie groups Quotient rings Torsion theory (Algebra) Geometrische Invariantentheorie (DE-588)4156712-2 gnd rswk-swf Gruppenoperation (DE-588)4158467-3 gnd rswk-swf Algebraische Gruppe (DE-588)4001164-1 gnd rswk-swf Algebraische Gruppe (DE-588)4001164-1 s Gruppenoperation (DE-588)4158467-3 s Geometrische Invariantentheorie (DE-588)4156712-2 s DE-604 Carrell, James B. Verfasser aut McGovern, William M. 1959- Verfasser (DE-588)124769179 aut Białynicki-Birula, Andrzej aut Quotients by actions of groups Carrell, James B. aut Torus actions and cohomology Invariant theory and algebraic transformation groups Encyclopaedia of mathematical sciences 2 (DE-604)BV014336202 2 Encyclopaedia of mathematical sciences 131 (DE-604)BV024126459 131 SWB Datenaustausch application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=009834664&sequence=000001&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis
spellingShingle	Białynicki-Birula, Andrzej Carrell, James B. McGovern, William M. 1959- Białynicki-Birula, Andrzej Carrell, James B. Algebraic quotients Encyclopaedia of mathematical sciences Geometria algébrica larpcal Homologie Lie, Algèbres de Lie, Groupes de Teoria geométrica de invariantes larpcal Torsion, Théorie de la (Algèbre) Homology theory Lie algebras Lie groups Quotient rings Torsion theory (Algebra) Geometrische Invariantentheorie (DE-588)4156712-2 gnd Gruppenoperation (DE-588)4158467-3 gnd Algebraische Gruppe (DE-588)4001164-1 gnd
subject_GND	(DE-588)4156712-2 (DE-588)4158467-3 (DE-588)4001164-1
title	Algebraic quotients
title_alt	Nebent.: Quotients by actions of groups Quotients by actions of groups Torus actions and cohomology
title_auth	Algebraic quotients
title_exact_search	Algebraic quotients
title_full	Algebraic quotients A. Białynicki-Birula ; J. B. Carrell ; W. M. McGovern
title_fullStr	Algebraic quotients A. Białynicki-Birula ; J. B. Carrell ; W. M. McGovern
title_full_unstemmed	Algebraic quotients A. Białynicki-Birula ; J. B. Carrell ; W. M. McGovern
title_short	Algebraic quotients
title_sort	algebraic quotients
topic	Geometria algébrica larpcal Homologie Lie, Algèbres de Lie, Groupes de Teoria geométrica de invariantes larpcal Torsion, Théorie de la (Algèbre) Homology theory Lie algebras Lie groups Quotient rings Torsion theory (Algebra) Geometrische Invariantentheorie (DE-588)4156712-2 gnd Gruppenoperation (DE-588)4158467-3 gnd Algebraische Gruppe (DE-588)4001164-1 gnd
topic_facet	Geometria algébrica Homologie Lie, Algèbres de Lie, Groupes de Teoria geométrica de invariantes Torsion, Théorie de la (Algèbre) Homology theory Lie algebras Lie groups Quotient rings Torsion theory (Algebra) Geometrische Invariantentheorie Gruppenoperation Algebraische Gruppe
url	http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=009834664&sequence=000001&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA
volume_link	(DE-604)BV014336202 (DE-604)BV024126459
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