Coxeter groups and Hopf algebras:
Gespeichert in:
| Hauptverfasser: | , |
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| Format: | Buch |
| Sprache: | English |
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Providence, Rhode Island
American Mathematical Society
[2006]
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| Schriftenreihe: | Fields Institute monographs / Fields Institute for Research in Mathematical Sciences
23 |
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| Online-Zugang: | Inhaltsverzeichnis |
| Beschreibung: | xvi, 181 Seiten Illustrationen, Diagramme |
| ISBN: | 0821839071 9780821853542 |
Internformat
MARC
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| 100 | 1 | |a Aguiar, Marcelo |d 1968- |e Verfasser |0 (DE-588)143265873 |4 aut | |
| 245 | 1 | 0 | |a Coxeter groups and Hopf algebras |c Marcelo Aguia, Swapneel Mahajan |
| 264 | 1 | |a Providence, Rhode Island |b American Mathematical Society |c [2006] | |
| 264 | 4 | |c © 2006 | |
| 300 | |a xvi, 181 Seiten |b Illustrationen, Diagramme | ||
| 336 | |b txt |2 rdacontent | ||
| 337 | |b n |2 rdamedia | ||
| 338 | |b nc |2 rdacarrier | ||
| 490 | 1 | |a Fields Institute monographs / Fields Institute for Research in Mathematical Sciences |v 23 | |
| 650 | 4 | |a Coxeter, Groupes de | |
| 650 | 4 | |a Hopf, Algèbres de | |
| 650 | 7 | |a Hopf-algebra's |2 gtt | |
| 650 | 4 | |a Hopf algebras | |
| 650 | 4 | |a Coxeter groups | |
| 650 | 0 | 7 | |a Hopf-Algebra |0 (DE-588)4160646-2 |2 gnd |9 rswk-swf |
| 650 | 0 | 7 | |a Coxeter-Gruppe |0 (DE-588)4261522-7 |2 gnd |9 rswk-swf |
| 689 | 0 | 0 | |a Hopf-Algebra |0 (DE-588)4160646-2 |D s |
| 689 | 0 | 1 | |a Coxeter-Gruppe |0 (DE-588)4261522-7 |D s |
| 689 | 0 | |5 DE-604 | |
| 700 | 1 | |a Mahajan, Swapneel |d 1974- |e Verfasser |0 (DE-588)143265970 |4 aut | |
| 776 | 0 | 8 | |i Erscheint auch als |n Online-Ausgabe |z 978-1-4704-1787-1 |
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| 943 | 1 | |a oai:aleph.bib-bvb.de:BVB01-015446253 | |
Datensatz im Suchindex
| _version_ | 1807864676704321536 |
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| adam_text |
FIELDS INSTITUTE MONOGRAPHS THE FIELDS INSTITUTE FOR RESEARCH IN
MATHEMATICAL SCIENCES COXETER GROUPS AND HOPF ALGEBRAS MARCELO AGUIAR
SWAPNEEL MAHAJAN AMERICAN MATHEMATICAL SOCIETY PROVIDENCE, RHODE ISLAND
CONTENTS LIST OF TABLES XI LIST OF FIGURES XIII PREFACE XV 0.1. THE
FIRST PART: CHAPTERS 1-3 XV 0.2. THE SECOND PART: CHAPTERS 4-8 XV 0.3.
FUTURE WORK XVI 0.4. ACKNOWLEDGEMENTS XVI 0.5. NOTATION XVI CHAPTER 1.
COXETER GROUPS 1 1.1. REGULAR CELL COMPLEXES AND SIMPLICIAL COMPLEXES 1
1.1.1. GATE PROPERTY 1 1.1.2. LINK AND JOIN ^ 2 1.2. HYPERPLANE
ARRANGEMENTS 2 1.2.1. FACES 2 1.2.2. FLATS 3 1.2.3. SPHERICAL PICTURE 4
1.2.4. GATE PROPERTY AND OTHER FACTS 4 1.3. REFLECTION ARRANGEMENTS 4
1.3.1. FINITE REFLECTION GROUPS 5 1.3.2. TYPES OF FACES 5 1.3.3. THE
COXETER DIAGRAM 5 1.3.4. THE DISTANCE MAP 6 1.3.5. THE BRUHAT ORDER 6
1.3.6. THE DESCENT ALGEBRA: A GEOMETRIC APPROACH 7 1.3.7. LINK AND JOIN
7 1.4. THE COXETER GROUP OF TYPE A N -\ 8 1.4.1. THE BRAID ARRANGEMENT 8
1.4.2. TYPES OF FACES 9 1.4.3. SET COMPOSITIONS AND PARTITIONS 9 1.4.4.
THE BRUHAT ORDER 10 CHAPTER 2. LEFT REGULAR BANDS 13 2.1. WHY LRBS? 13
2.2. FACES AND FLATS 14 2.2.1. FACES 14 2.2.2. FLATS 14 VI CONTENTS
2.2.3. CHAMBERS 14 2.2.4. EXAMPLES 15 2.3. POINTED FACES AND LUNES 15
2.3.1. POINTED FACES 15 2.3.2. LUNES 15 2.3.3. THE RELATION OF Q AND Z
WITH E AND L 16 2.3.4. LUNAR REGIONS 16 2.3.5. EXAMPLES 17 2.4. LINK AND
JOIN OF LRBS 19 2.4.1. SUBLRB AND QUOTIENT LRB 19 2.4.2. PRODUCT OF LRBS
19 2.5. BILINEAR FORMS RELATED TO AN LRB 19 2.5.1. THE BILINEAR FORM ON
KQ 20 2.5.2. THE PAIRING BETWEEN KQ AND KE 21 2.5.3. THE BILINEAR FORM
ON KE 21 2.5.4. THE BILINEAR FORM ON KL 22 2.5.5. THE NONDEGENERACY OF
THE FORM ON KL 22 2.6. BILINEAR FORMS RELATED TO A COXETER GROUP 24
2.6.1. THE BILINEAR FORM ON (KE) W 24 2.6.2. THE BILINEAR FORM ON (KL) 1
^ AND ITS NONDEGENERACY 25 2.7. PROJECTION POSETS 26 2.7.1. DEFINITION
AND EXAMPLES 26 2.7.2. ELEMENTARY FACTS 27 CHAPTER 3. HOPF ALGEBRAS 31
3.1. HOPF ALGEBRAS 31 3.1.1. COFREE GRADED COALGEBRAS 31 3.1.2. THE
CORADICAL FILTRATION 32 3.1.3. ANTIPODE 32 3.2. HOPF ALGEBRAS: EXAMPLES
33 3.2.1. THE HOPF ALGEBRA A 33 3.2.2. THE HOPF ALGEBRA QA 35 3.2.3. THE
HOPF ALGEBRA NA 36 3.2.4. THE DUALITY BETWEEN QA AND NA 37 CHAPTER 4. A
BRIEF OVERVIEW 39 4.1. ABSTRACT: CHAPTER 5 39 4.2. ABSTRACT: CHAPTER 6
40 4.3. ABSTRACT: CHAPTERS 7 AND 8 41 CHAPTER 5. THE DESCENT THEORY FOR
COXETER GROUPS 43 5.1. INTRODUCTION 43 5.1.1. THE FIRST PART: SECTIONS
5.2-5.5 43 5.1.2. THE SECOND PART: SECTIONS 5.6-5.7 44 5.2. THE DESCENT
THEORY FOR COXETER GROUPS 45 5.2.1. PRELIMINARIES 45 5.2.2. SUMMARY 45
5.2.3. THE POSETS Z AND L 46 5.2.4. THE PARTIAL ORDERS ON C X C AND Q 46
5.2.5. THE MAP ROAD 48 CONTENTS VII 5.2.6. THE MAP GROAD 49 5.2.7. THE
MAP 0 50 5.2.8. CONNECTION AMONG THE THREE MAPS 51 5.3. THE COINVARIANT
DESCENT THEORY FOR COXETER GROUPS 52 5.3.1. THE MAP DES 52 5.3.2. THE
MAP GDES 53 5.3.3. THE MAP (9 53 5.3.4. CONNECTION AMONG THE THREE MAPS
54 5.3.5. SHUFFLES 55 5.3.6. SETS RELATED TO THE PRODUCT IN THE M BASIS
OF SA 57 5.4. THE EXAMPLE OF TYPE A N -I 58 5.4.1. THE POSETS S* AND L N
59 5.4.2. THE POSETS Q" AND Z_^ 59 5.4.3. THE QUOTIENT POSETS Q AND L 60
5.4.4. THE MAPS ROAD, GROAD AND 0 60 5.4.5. THE MAPS DES, GDES AND 6 6 1
5.4.6. SHUFFLES 62 5.5. THE TOY EXAMPLE OF TYPE A* {N ~ 1] 62 5.5.1. THE
POSETS E * AND L N 62 5.5.2. THE POSETS Q * AND Z" 63 5.5.3. THE
QUOTIENT POSETS Q * AND L * 63 5.5.4. THE MAPS DES, GDES AND 6 64 5.5.5.
THE MAPS DES, GDES AND 9 64 5.6. THE COMMUTATIVE DIAGRAM (5.8) 64 5.6.1.
THE OBJECTS IN DIAGRAM (5.8) 65 5.6.2. THE MAPS S, 6 AND ROAD 66 5.6.3.
THE BILINEAR FORM ON KQ 67 5.6.4. THE TOP HALF OF DIAGRAM (5.8) 68
5.6.5. THE MAPS SUPP, LUNE AND BASE* ^ 68 5.6.6. THE DUAL MAPS SUPP*,
LUNE* AND BASE 69 5.6.7. THE MAPS $ AND T 69 5.6.8. THE BOTTOM HALF OF
DIAGRAM (5.8) 69 5.6.9. THE ALGEBRA KL 70 5.7. THE COINVARIANT
COMMUTATIVE DIAGRAM (5.17) 71 5.7.1. THE OBJECTS IN DIAGRAM (5.17) 72
5.7.2. THE MAPS FROM INVARIANTS 73 5.7.3. THE MAPS TO COINVARIANTS 75
5.7.4. THE MAPS IN DIAGRAM (5.17) 76 5.7.5. THE ALGEBRA KL 77 5.7.6. A
DIFFERENT VIEWPOINT RELATING DIAGRAMS (5.8) AND (5.17) 78 CHAPTER 6. THE
CONSTRUCTION OF HOPF ALGEBRAS 81 6.1. INTRODUCTION 81 6.1.1. A DIAGRAM
OF VECTOR SPACES FOR AN LRB 81 6.1.2. A DIAGRAM OF COALGEBRAS AND
ALGEBRAS FOR A FAMILY OF LRBS 82 6.1.3. THE EXAMPLE OF TYPE A 83 6.2.
THE HOPF ALGEBRAS OF TYPE A " 85 6.2.1. SUMMARY 85 6.2.2. THE STRUCTURE
OF THE HOPF ALGEBRAS OF TYPE A 86 I CONTENTS 6.2.3. SET COMPOSITIONS 86
6.2.4. THE HOPF ALGEBRA PN 88 6.2.5. THE HOPF ALGEBRA MN 89 6.2.6.
NESTED SET COMPOSITIONS 89 6.2.7. THE HOPF ALGEBRA QN 90 6.2.8. THE HOPF
ALGEBRA MI 91 6.2.9. SET PARTITIONS 91 6.2.10. THE HOPF ALGEBRA N L * 91
6.2.11. THE HOPF ALGEBRA U L 92 6.2.12. NESTED SET PARTITIONS 92 6.2.13.
THE HOPF ALGEBRA N Z - 93 6.2.14. THE HOPF ALGEBRA IL Z 93 6.2.15. THE
HOPF ALGEBRA SN 93 6.2.16. THE HOPF ALGEBRA RN 94 6.3. THE COALGEBRA
AXIOMS AND EXAMPLES 94 6.3.1. THE COALGEBRA AXIOMS 94 6.3.2. THE WARM-UP
EXAMPLE OF COMPOSITIONS 96 6.3.3. THE MOTIVATING EXAMPLE OF TYPE A N -I
97 6.3.4. THE EXAMPLE OF TYPE A* {N ~ 1] 101 6.4. FROM COALGEBRA AXIOMS
TO COALGEBRAS 102 6.4.1. THE COPRODUCTS 102 6.4.2. COASSOCIATIVITY OF
THE COPRODUCTS 103 6.4.3. USEFUL RESULTS FOR COASSOCIATIVITY 103 6.5.
CONSTRUCTION OF COALGEBRAS 105 6.5.1. EXAMPLES 105 6.5.2. THE COPRODUCTS
AND LOCAL AND GLOBAL VERTICES 106 6.5.3. THE COALGEBRA V 106 6.5.4. THE
COALGEBRA M 108 6.5.5. THE COALGEBRA Q ^ 110 6.5.6. THE COALGEBRA N ^
111 6.5.7. THE COALGEBRA S 112 6.5.8. THE COALGEBRA 1Z 114 6.5.9. THE
MAPS ROAD : S - Q AND 0 : M - 11 114 6.5.10. THE COALGEBRAS A Z , AC,
AZ* AND AC* 115 6.6. THE ALGEBRA AXIOMS AND EXAMPLES 117 6.6.1. THE
ALGEBRA AXIOMS 117 6.6.2. THE WARM-UP EXAMPLE OF COMPOSITIONS 119 6.6.3.
THE MOTIVATING EXAMPLE OF TYPE A N -\ 120 6.6.4. THE EXAMPLE OF TYPE A*
{N ~ L) 121 6.7. FROM ALGEBRA AXIOMS TO ALGEBRAS 122 6.7.1. THE PRODUCTS
122 6.7.2. ASSOCIATIVITY OF THE PRODUCTS 122 6.7.3. USEFUL RESULTS FOR
ASSOCIATIVITY 123 6.8. CONSTRUCTION OF ALGEBRAS 123 6.8.1. EXAMPLES 124
6.8.2. THE ALGEBRA V 124 6.8.3. THE ALGEBRA M 126 6.8.4. THE ALGEBRA Q
127 CONTENTS IX 6.8.5. THE ALGEBRA AF 128 6.8.6. THE ALGEBRA S 128
6.8.7. THE ALGEBRA U 129 6.8.8. THE MAPS ROAD : S - Q AND 0 : M - H
130 6.8.9. THE ALGEBRAS A Z , A C , A Z * AND AC* 131 CHAPTER 7. THE
HOPF ALGEBRA OF PAIRS OF PERMUTATIONS 133 7.1. INTRODUCTION 133 7.1.1.
THE BASIC SETUP 133 7.1.2. THE MAIN RESULT 133 7.1.3. THE HOPF ALGEBRAS
RN AND RA 134 7.1.4. THREE PARTIAL ORDERS ON C" X C N 135 7.1.5. THE
DIFFERENT BASES OF SN AND SA 135 7.1.6. THE PROOF METHOD AND THE
ORGANIZATION OF THE CHAPTER 136 7.2. THE HOPF ALGEBRA SN 137 7.2.1.
PRELIMINARY DEFINITIONS 137 7.2.2. COMBINATORIAL DEFINITION 138 7.2.3.
THE BREAK AND JOIN OPERATIONS 138 7.2.4. GEOMETRIC DEFINITION 139 7.2.5.
THE HOPF ALGEBRA SA 139 7.3. THE HOPF ALGEBRA SN IN THE M BASI S 141
7.3.1. A PRELIMINARY RESULT 141 7.3.2. COPRODUCT IN THE M BASIS 141
7.3.3. PRODUCT IN THE M BASIS 143 7.3.4. THE SWITCH MAP ON THE M BASIS
145 7.4. THE HOPF ALGEBRA SIT IN THE S BASIS 145 7.4.1. TWO PRELIMINARY
RESULTS 146 7.4.2. COPRODUCT IN THE S BASIS 147 7.4.3. PRODUCT IN THE
STASIS 150 7.5. THE HOPF ALGEBRA RN IN THE H BASIS 150 7.5.1. COPRODUCT
IN THE H BASIS 151 7.5.2. PRODUCT IN THE H BASIS 152 7.5.3. THE SWITCH
MAP ON THE H BASIS 153 CHAPTER 8. THE HOPF ALGEBRA OF POINTED FACES 155
8.1. INTRODUCTION 155 8.1.1. THE BASIC SETUP 155 8.1.2. COFREENESS 155
8.1.3. THREE PARTIAL ORDERS ON Q N 156 8.1.4. THE DIFFERENT BASES OF QN
156 8.1.5. THE CONNECTION BETWEEN SN AND QN 157 8.2. THE HOPF ALGEBRA QN
157 8.2.1. GEOMETRIC DEFINITION 157 8.2.2. COMBINATORIAL DEFINITION 159
8.3. THE HOPF ALGEBRA PN ' 161 8.4. THE HOPF ALGEBRA QA OF
QUASI-SYMMETRIC FUNCTIONS 162 BIBLIOGRAPHY 165 AUTHOR INDEX 171 X
CONTENTS NOTATION INDEX 173 SUBJECT INDEX 177 |
| adam_txt |
FIELDS INSTITUTE MONOGRAPHS THE FIELDS INSTITUTE FOR RESEARCH IN
MATHEMATICAL SCIENCES COXETER GROUPS AND HOPF ALGEBRAS MARCELO AGUIAR
SWAPNEEL MAHAJAN AMERICAN MATHEMATICAL SOCIETY PROVIDENCE, RHODE ISLAND
CONTENTS LIST OF TABLES XI LIST OF FIGURES XIII PREFACE XV 0.1. THE
FIRST PART: CHAPTERS 1-3 XV 0.2. THE SECOND PART: CHAPTERS 4-8 XV 0.3.
FUTURE WORK XVI 0.4. ACKNOWLEDGEMENTS XVI 0.5. NOTATION XVI CHAPTER 1.
COXETER GROUPS 1 1.1. REGULAR CELL COMPLEXES AND SIMPLICIAL COMPLEXES 1
1.1.1. GATE PROPERTY 1 1.1.2. LINK AND JOIN ^ 2 1.2. HYPERPLANE
ARRANGEMENTS 2 1.2.1. FACES 2 1.2.2. FLATS 3 1.2.3. SPHERICAL PICTURE 4
1.2.4. GATE PROPERTY AND OTHER FACTS 4 1.3. REFLECTION ARRANGEMENTS 4
1.3.1. FINITE REFLECTION GROUPS 5 1.3.2. TYPES OF FACES 5 1.3.3. THE
COXETER DIAGRAM 5 1.3.4. THE DISTANCE MAP 6 1.3.5. THE BRUHAT ORDER 6
1.3.6. THE DESCENT ALGEBRA: A GEOMETRIC APPROACH 7 1.3.7. LINK AND JOIN
7 1.4. THE COXETER GROUP OF TYPE A N -\ 8 1.4.1. THE BRAID ARRANGEMENT 8
1.4.2. TYPES OF FACES 9 1.4.3. SET COMPOSITIONS AND PARTITIONS 9 1.4.4.
THE BRUHAT ORDER 10 CHAPTER 2. LEFT REGULAR BANDS 13 2.1. WHY LRBS? 13
2.2. FACES AND FLATS 14 2.2.1. FACES 14 2.2.2. FLATS 14 VI CONTENTS
2.2.3. CHAMBERS 14 2.2.4. EXAMPLES 15 2.3. POINTED FACES AND LUNES 15
2.3.1. POINTED FACES 15 2.3.2. LUNES 15 2.3.3. THE RELATION OF Q AND Z
WITH E AND L 16 2.3.4. LUNAR REGIONS 16 2.3.5. EXAMPLES 17 2.4. LINK AND
JOIN OF LRBS 19 2.4.1. SUBLRB AND QUOTIENT LRB 19 2.4.2. PRODUCT OF LRBS
19 2.5. BILINEAR FORMS RELATED TO AN LRB 19 2.5.1. THE BILINEAR FORM ON
KQ 20 2.5.2. THE PAIRING BETWEEN KQ AND KE 21 2.5.3. THE BILINEAR FORM
ON KE 21 2.5.4. THE BILINEAR FORM ON KL 22 2.5.5. THE NONDEGENERACY OF
THE FORM ON KL 22 2.6. BILINEAR FORMS RELATED TO A COXETER GROUP 24
2.6.1. THE BILINEAR FORM ON (KE) W 24 2.6.2. THE BILINEAR FORM ON (KL) 1
^ AND ITS NONDEGENERACY 25 2.7. PROJECTION POSETS 26 2.7.1. DEFINITION
AND EXAMPLES 26 2.7.2. ELEMENTARY FACTS 27 CHAPTER 3. HOPF ALGEBRAS 31
3.1. HOPF ALGEBRAS 31 3.1.1. COFREE GRADED COALGEBRAS 31 3.1.2. THE
CORADICAL FILTRATION 32 3.1.3. ANTIPODE 32 3.2. HOPF ALGEBRAS: EXAMPLES
33 3.2.1. THE HOPF ALGEBRA A 33 3.2.2. THE HOPF ALGEBRA QA 35 3.2.3. THE
HOPF ALGEBRA NA 36 3.2.4. THE DUALITY BETWEEN QA AND NA 37 CHAPTER 4. A
BRIEF OVERVIEW 39 4.1. ABSTRACT: CHAPTER 5 39 4.2. ABSTRACT: CHAPTER 6
40 4.3. ABSTRACT: CHAPTERS 7 AND 8 41 CHAPTER 5. THE DESCENT THEORY FOR
COXETER GROUPS 43 5.1. INTRODUCTION 43 5.1.1. THE FIRST PART: SECTIONS
5.2-5.5 43 5.1.2. THE SECOND PART: SECTIONS 5.6-5.7 44 5.2. THE DESCENT
THEORY FOR COXETER GROUPS 45 5.2.1. PRELIMINARIES 45 5.2.2. SUMMARY 45
5.2.3. THE POSETS Z AND L 46 5.2.4. THE PARTIAL ORDERS ON C X C AND Q 46
5.2.5. THE MAP ROAD 48 CONTENTS VII 5.2.6. THE MAP GROAD 49 5.2.7. THE
MAP 0 50 5.2.8. CONNECTION AMONG THE THREE MAPS 51 5.3. THE COINVARIANT
DESCENT THEORY FOR COXETER GROUPS 52 5.3.1. THE MAP DES 52 5.3.2. THE
MAP GDES 53 5.3.3. THE MAP (9 53 5.3.4. CONNECTION AMONG THE THREE MAPS
54 5.3.5. SHUFFLES 55 5.3.6. SETS RELATED TO THE PRODUCT IN THE M BASIS
OF SA 57 5.4. THE EXAMPLE OF TYPE A N -I 58 5.4.1. THE POSETS S* AND L N
59 5.4.2. THE POSETS Q" AND Z_^ 59 5.4.3. THE QUOTIENT POSETS Q AND L 60
5.4.4. THE MAPS ROAD, GROAD AND 0 60 5.4.5. THE MAPS DES, GDES AND 6 6 1
5.4.6. SHUFFLES 62 5.5. THE TOY EXAMPLE OF TYPE A* {N ~ 1] 62 5.5.1. THE
POSETS E * AND L N 62 5.5.2. THE POSETS Q * AND Z" 63 5.5.3. THE
QUOTIENT POSETS Q * AND L * 63 5.5.4. THE MAPS DES, GDES AND 6 64 5.5.5.
THE MAPS DES, GDES AND 9 64 5.6. THE COMMUTATIVE DIAGRAM (5.8) 64 5.6.1.
THE OBJECTS IN DIAGRAM (5.8) 65 5.6.2. THE MAPS S, 6 AND ROAD 66 5.6.3.
THE BILINEAR FORM ON KQ 67 5.6.4. THE TOP HALF OF DIAGRAM (5.8) 68
5.6.5. THE MAPS SUPP, LUNE AND BASE* ^ 68 5.6.6. THE DUAL MAPS SUPP*,
LUNE* AND BASE 69 5.6.7. THE MAPS $ AND T 69 5.6.8. THE BOTTOM HALF OF
DIAGRAM (5.8) 69 5.6.9. THE ALGEBRA KL 70 5.7. THE COINVARIANT
COMMUTATIVE DIAGRAM (5.17) 71 5.7.1. THE OBJECTS IN DIAGRAM (5.17) 72
5.7.2. THE MAPS FROM INVARIANTS 73 5.7.3. THE MAPS TO COINVARIANTS 75
5.7.4. THE MAPS IN DIAGRAM (5.17) 76 5.7.5. THE ALGEBRA KL 77 5.7.6. A
DIFFERENT VIEWPOINT RELATING DIAGRAMS (5.8) AND (5.17) 78 CHAPTER 6. THE
CONSTRUCTION OF HOPF ALGEBRAS 81 6.1. INTRODUCTION 81 6.1.1. A DIAGRAM
OF VECTOR SPACES FOR AN LRB 81 6.1.2. A DIAGRAM OF COALGEBRAS AND
ALGEBRAS FOR A FAMILY OF LRBS 82 6.1.3. THE EXAMPLE OF TYPE A 83 6.2.
THE HOPF ALGEBRAS OF TYPE A " 85 6.2.1. SUMMARY 85 6.2.2. THE STRUCTURE
OF THE HOPF ALGEBRAS OF TYPE A 86 I CONTENTS 6.2.3. SET COMPOSITIONS 86
6.2.4. THE HOPF ALGEBRA PN 88 6.2.5. THE HOPF ALGEBRA MN 89 6.2.6.
NESTED SET COMPOSITIONS 89 6.2.7. THE HOPF ALGEBRA QN 90 6.2.8. THE HOPF
ALGEBRA MI 91 6.2.9. SET PARTITIONS 91 6.2.10. THE HOPF ALGEBRA N L * 91
6.2.11. THE HOPF ALGEBRA U L 92 6.2.12. NESTED SET PARTITIONS 92 6.2.13.
THE HOPF ALGEBRA N Z - 93 6.2.14. THE HOPF ALGEBRA IL Z 93 6.2.15. THE
HOPF ALGEBRA SN 93 6.2.16. THE HOPF ALGEBRA RN 94 6.3. THE COALGEBRA
AXIOMS AND EXAMPLES 94 6.3.1. THE COALGEBRA AXIOMS 94 6.3.2. THE WARM-UP
EXAMPLE OF COMPOSITIONS 96 6.3.3. THE MOTIVATING EXAMPLE OF TYPE A N -I
97 6.3.4. THE EXAMPLE OF TYPE A* {N ~ 1] 101 6.4. FROM COALGEBRA AXIOMS
TO COALGEBRAS 102 6.4.1. THE COPRODUCTS 102 6.4.2. COASSOCIATIVITY OF
THE COPRODUCTS 103 6.4.3. USEFUL RESULTS FOR COASSOCIATIVITY 103 6.5.
CONSTRUCTION OF COALGEBRAS 105 6.5.1. EXAMPLES 105 6.5.2. THE COPRODUCTS
AND LOCAL AND GLOBAL VERTICES 106 6.5.3. THE COALGEBRA V 106 6.5.4. THE
COALGEBRA M 108 6.5.5. THE COALGEBRA Q ^ 110 6.5.6. THE COALGEBRA N ^
111 6.5.7. THE COALGEBRA S 112 6.5.8. THE COALGEBRA 1Z 114 6.5.9. THE
MAPS ROAD : S - Q AND 0 : M - 11 114 6.5.10. THE COALGEBRAS A Z , AC,
AZ* AND AC* 115 6.6. THE ALGEBRA AXIOMS AND EXAMPLES 117 6.6.1. THE
ALGEBRA AXIOMS 117 6.6.2. THE WARM-UP EXAMPLE OF COMPOSITIONS 119 6.6.3.
THE MOTIVATING EXAMPLE OF TYPE A N -\ 120 6.6.4. THE EXAMPLE OF TYPE A*
{N ~ L) 121 6.7. FROM ALGEBRA AXIOMS TO ALGEBRAS 122 6.7.1. THE PRODUCTS
122 6.7.2. ASSOCIATIVITY OF THE PRODUCTS 122 6.7.3. USEFUL RESULTS FOR
ASSOCIATIVITY 123 6.8. CONSTRUCTION OF ALGEBRAS 123 6.8.1. EXAMPLES 124
6.8.2. THE ALGEBRA V 124 6.8.3. THE ALGEBRA M 126 6.8.4. THE ALGEBRA Q
127 CONTENTS IX 6.8.5. THE ALGEBRA AF 128 6.8.6. THE ALGEBRA S 128
6.8.7. THE ALGEBRA U 129 6.8.8. THE MAPS ROAD : S - Q AND 0 : M - H
130 6.8.9. THE ALGEBRAS A Z , A C , A Z * AND AC* 131 CHAPTER 7. THE
HOPF ALGEBRA OF PAIRS OF PERMUTATIONS 133 7.1. INTRODUCTION 133 7.1.1.
THE BASIC SETUP 133 7.1.2. THE MAIN RESULT 133 7.1.3. THE HOPF ALGEBRAS
RN AND RA 134 7.1.4. THREE PARTIAL ORDERS ON C" X C N 135 7.1.5. THE
DIFFERENT BASES OF SN AND SA 135 7.1.6. THE PROOF METHOD AND THE
ORGANIZATION OF THE CHAPTER 136 7.2. THE HOPF ALGEBRA SN 137 7.2.1.
PRELIMINARY DEFINITIONS 137 7.2.2. COMBINATORIAL DEFINITION 138 7.2.3.
THE BREAK AND JOIN OPERATIONS 138 7.2.4. GEOMETRIC DEFINITION 139 7.2.5.
THE HOPF ALGEBRA SA 139 7.3. THE HOPF ALGEBRA SN IN THE M BASI S 141
7.3.1. A PRELIMINARY RESULT 141 7.3.2. COPRODUCT IN THE M BASIS 141
7.3.3. PRODUCT IN THE M BASIS 143 7.3.4. THE SWITCH MAP ON THE M BASIS
145 7.4. THE HOPF ALGEBRA SIT IN THE S BASIS 145 7.4.1. TWO PRELIMINARY
RESULTS 146 7.4.2. COPRODUCT IN THE S BASIS 147 7.4.3. PRODUCT IN THE
STASIS 150 7.5. THE HOPF ALGEBRA RN IN THE H BASIS 150 7.5.1. COPRODUCT
IN THE H BASIS 151 7.5.2. PRODUCT IN THE H BASIS 152 7.5.3. THE SWITCH
MAP ON THE H BASIS 153 CHAPTER 8. THE HOPF ALGEBRA OF POINTED FACES 155
8.1. INTRODUCTION 155 8.1.1. THE BASIC SETUP 155 8.1.2. COFREENESS 155
8.1.3. THREE PARTIAL ORDERS ON Q N 156 8.1.4. THE DIFFERENT BASES OF QN
156 8.1.5. THE CONNECTION BETWEEN SN AND QN 157 8.2. THE HOPF ALGEBRA QN
157 8.2.1. GEOMETRIC DEFINITION 157 8.2.2. COMBINATORIAL DEFINITION 159
8.3. THE HOPF ALGEBRA PN ' 161 8.4. THE HOPF ALGEBRA QA OF
QUASI-SYMMETRIC FUNCTIONS 162 BIBLIOGRAPHY 165 AUTHOR INDEX 171 X
CONTENTS NOTATION INDEX 173 SUBJECT INDEX 177 |
| any_adam_object | 1 |
| any_adam_object_boolean | 1 |
| author | Aguiar, Marcelo 1968- Mahajan, Swapneel 1974- |
| author_GND | (DE-588)143265873 (DE-588)143265970 |
| author_facet | Aguiar, Marcelo 1968- Mahajan, Swapneel 1974- |
| author_role | aut aut |
| author_sort | Aguiar, Marcelo 1968- |
| author_variant | m a ma s m sm |
| building | Verbundindex |
| bvnumber | BV022235234 |
| callnumber-first | Q - Science |
| callnumber-label | QA613 |
| callnumber-raw | QA613.8 |
| callnumber-search | QA613.8 |
| callnumber-sort | QA 3613.8 |
| callnumber-subject | QA - Mathematics |
| classification_rvk | SK 230 SK 260 |
| ctrlnum | (OCoLC)65165510 (DE-599)BVBBV022235234 |
| dewey-full | 512/.55 |
| dewey-hundreds | 500 - Natural sciences and mathematics |
| dewey-ones | 512 - Algebra |
| dewey-raw | 512/.55 |
| dewey-search | 512/.55 |
| dewey-sort | 3512 255 |
| dewey-tens | 510 - Mathematics |
| discipline | Mathematik |
| discipline_str_mv | Mathematik |
| format | Book |
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| id | DE-604.BV022235234 |
| illustrated | Illustrated |
| index_date | 2024-07-02T16:33:53Z |
| indexdate | 2024-08-20T00:35:09Z |
| institution | BVB |
| isbn | 0821839071 9780821853542 |
| language | English |
| lccn | 2006044456 |
| oai_aleph_id | oai:aleph.bib-bvb.de:BVB01-015446253 |
| oclc_num | 65165510 |
| open_access_boolean | |
| owner | DE-703 DE-11 DE-188 |
| owner_facet | DE-703 DE-11 DE-188 |
| physical | xvi, 181 Seiten Illustrationen, Diagramme |
| publishDate | 2006 |
| publishDateSearch | 2006 |
| publishDateSort | 2006 |
| publisher | American Mathematical Society |
| record_format | marc |
| series2 | Fields Institute monographs / Fields Institute for Research in Mathematical Sciences |
| spelling | Aguiar, Marcelo 1968- Verfasser (DE-588)143265873 aut Coxeter groups and Hopf algebras Marcelo Aguia, Swapneel Mahajan Providence, Rhode Island American Mathematical Society [2006] © 2006 xvi, 181 Seiten Illustrationen, Diagramme txt rdacontent n rdamedia nc rdacarrier Fields Institute monographs / Fields Institute for Research in Mathematical Sciences 23 Coxeter, Groupes de Hopf, Algèbres de Hopf-algebra's gtt Hopf algebras Coxeter groups Hopf-Algebra (DE-588)4160646-2 gnd rswk-swf Coxeter-Gruppe (DE-588)4261522-7 gnd rswk-swf Hopf-Algebra (DE-588)4160646-2 s Coxeter-Gruppe (DE-588)4261522-7 s DE-604 Mahajan, Swapneel 1974- Verfasser (DE-588)143265970 aut Erscheint auch als Online-Ausgabe 978-1-4704-1787-1 Fields Institute for Research in Mathematical Sciences Fields Institute monographs 23 (DE-604)BV009737926 23 GBV Datenaustausch application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=015446253&sequence=000001&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Aguiar, Marcelo 1968- Mahajan, Swapneel 1974- Coxeter groups and Hopf algebras Coxeter, Groupes de Hopf, Algèbres de Hopf-algebra's gtt Hopf algebras Coxeter groups Hopf-Algebra (DE-588)4160646-2 gnd Coxeter-Gruppe (DE-588)4261522-7 gnd |
| subject_GND | (DE-588)4160646-2 (DE-588)4261522-7 |
| title | Coxeter groups and Hopf algebras |
| title_auth | Coxeter groups and Hopf algebras |
| title_exact_search | Coxeter groups and Hopf algebras |
| title_exact_search_txtP | Coxeter groups and Hopf algebras |
| title_full | Coxeter groups and Hopf algebras Marcelo Aguia, Swapneel Mahajan |
| title_fullStr | Coxeter groups and Hopf algebras Marcelo Aguia, Swapneel Mahajan |
| title_full_unstemmed | Coxeter groups and Hopf algebras Marcelo Aguia, Swapneel Mahajan |
| title_short | Coxeter groups and Hopf algebras |
| title_sort | coxeter groups and hopf algebras |
| topic | Coxeter, Groupes de Hopf, Algèbres de Hopf-algebra's gtt Hopf algebras Coxeter groups Hopf-Algebra (DE-588)4160646-2 gnd Coxeter-Gruppe (DE-588)4261522-7 gnd |
| topic_facet | Coxeter, Groupes de Hopf, Algèbres de Hopf-algebra's Hopf algebras Coxeter groups Hopf-Algebra Coxeter-Gruppe |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=015446253&sequence=000001&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
| volume_link | (DE-604)BV009737926 |
| work_keys_str_mv | AT aguiarmarcelo coxetergroupsandhopfalgebras AT mahajanswapneel coxetergroupsandhopfalgebras |