On the classification of hyperbolic root systems of rank three:
Gespeichert in:
| Beteilige Person: | |
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| Format: | Buch |
| Sprache: | Englisch |
| Veröffentlicht: |
Birmingham
Interperiodica
2000
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| Schriftenreihe: | Matematičeskij Institut Imeni V. A. Steklova <Moskva>: Proceedings of the Steklov Institute of Mathematics
Volume 230 = 2000,3 |
| Schlagwörter: | |
| Links: | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=009171502&sequence=000001&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
| Beschreibung: | Aus dem Russischen übersetzt |
| Umfang: | 241 Seite |
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| 245 | 1 | 0 | |a On the classification of hyperbolic root systems of rank three |c V. V. Nikulin |
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Datensatz im Suchindex
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|---|---|
| adam_text | Titel: On the classification of hyperbolic root systems of rank three
Autor: Nikulin, Vjačeslav V
Jahr: 2000
Contents
Volume 230, 2000
INTRODUCTION .................................................................................... 1
Part I. CLASSIFICATION OF MAXIMAL HYPERBOLIC ROOT SYSTEMS OF RANK
THREE OF ELLIPTIC OR PARABOLIC TYPES
Chapter 1. HYPERBOLIC REFLECTION GROUPS AND HYPERBOLIC ROOT SYSTEMS.........4
1.1. The Klein Model of a Hyperbolic Space ............................................... 4
1.2. Reflection Groups of Hyperbolic Lattices and Hyperbolic Root Systems ................ 6
1.3. Restricted Arithmetic Type and a Generalized Lattice Weyl Vector .................... 7
1.4. Reflective Hyperbolic Lattices......................................................... 7
Chapter 2. CLASSIFICATION OF ELLIPTICALLY OR PARABOLICALLY REFLECTIVE HY-
PERBOLIC LATTICES OF RANK THREE: FORMULATIONS...........................8
2.1. The Principle of Classification ........................................................ 8
2.2. Elementary Hyperbolic Lattices of Rank 3.............................................9
2.3. The Classification of Elliptically or Parabolically Reflective Elementary Hyperbolic
Lattices of Rank 3 ................................................................... 13
2.3.1. Notation (13). 2.3.2. Classification of elliptically or parabolically reflective
main hyperbolic lattices of rank three (14). 2.3.3. Classification of elliptically or
parabolically reflective non-main hyperbolic lattices of rank three with a square-
free determinant (15). 2.3.4. Classification of elliptically or parabolically reflective
elementary hyperbolic lattices of rank three (16)
Chapter 3. THE NUMBER h OF CLASSES OF CENTRAL SYMMETRIES OF MAIN HYPER-
BOLIC LATTICES OF RANK THREE .................................................. 17
3.1. Some Classical Results about Binary Positive Lattices ................................ 17
3.2. The Number h of Non-reflective Classes of Central Symmetries of Main Hyperbolic
Lattices of Rank 3 ................................................................... 22
Chapter 4. NARROW PARTS OF ELLIPTIC AND PARABOLIC CONVEX POLYGONS ON THE
HYPERBOLIC PLANE, TYPES OF POLYGONS. APPLICATION TO REFLECTIVE
LATTICES OF RANK THREE .......................................................... 27
4.1. Narrow Parts of Elliptic Convex Polygons on the Hyperbolic Plane ................... 27
4.2. Narrow Parts of Restricted Parabolic Convex Polygons on the Hyperbolic Plane ...... 43
4.3. Description of Narrow Parts of Fundamental Polygons M of Reflection Subgroups
W C V(S) of Elliptic and Parabolic Types for rkS = 3. Application to Elliptically
or Parabolically Reflective Hyperbolic Lattices of Rank Three ........................ 47
4.3.1. Matrices B of the narrow parts of type (II) (49). 4.3.2. Matrices B of the nar-
row parts of type (10) (50). 4.3.3. Matrices B of the narrow parts of type (III) (51).
4.3.4. Matrices B of the narrow parts of type (110) (52). 4.3.5. Matrices B of the
narrow parts of type (III) (53). 4.3.6. The global estimate of invariants of elliptically
or parabolically reflective hyperbolic lattices of rank 3 (54)
Chapter 5. CLASSIFICATION OF ELLIPTICALLY OR PARABOLICALLY REFLECTIVE HY-
PERBOLIC LATTICES OF RANK THREE: PROOFS...................................55
5.1. Proof of Basic Theorem 2.3.3 ........................................................ 55
5.2. Proof of Theorem 2.3.7 .............................................................. 63
Appendix 1.1. TABLES 1-3 ..........................................................................64
Table 1. The List of Elliptically or Parabolically Reflective Main Hyperbolic Lattices of
Rank Three.....................................................................64
Table 2. The List of Elliptically or Parabolically Reflective Odd Hyperbolic Lattices of
Rank Three with the Even Square-Free Determinant (i.e., Non-main) ............ 96
Table 3. The List of Main Hyperbolic Lattices of Rank Three with the Determinant d
100000 and h ..............................................................102
Appendix 1.2. PROGRAMS FOR THE GP/PARI CALCULATOR. ELLIPTIC AND PARABOLIC
TYPES .............................................................................. 107
Program 1: h2 .......................................................................... 107
Program 2: h3 .......................................................................... 109
Program 3: refh3 ....................................................................... 112
Program 4: fundll.gen.................................................................. 113
Program 5: fundlO.gen.................................................................. 113
Program 6: fund21.gen.................................................................. 114
Program 7: fund20.gen.................................................................. 115
Program 8: fund30.gen.................................................................. 11G
Program 9: fund20.main ................................................................ 117
Program 10: reflO.l ..................................................................... 119
Program 11: refl0.12 .................................................................... 122
Program 12: refl0.13 .................................................................... 123
Part II. CLASSIFICATION OF MAXIMAL HYPERBOLIC ROOT SYSTEMS OF RANK
THREE OF HYPERBOLIC TYPE
Chapter 6. CLASSIFICATION OF HYPERBOLICALLY REFLECTIVE HYPERBOLIC LAT-
TICES OF RANK THREE: FORMULATIONS..........................................125
Chapter 7. NARROW PARTS OF CONVEX POLYGONS OF RESTRICTED HYPERBOLIC
TYPE ON THE HYPERBOLIC PLANE. APPLICATION TO HYPERBOLICALLY
REFLECTIVE HYPERBOLIC LATTICES..............................................128
7.1. Some Formulae Involving the Cross-ratio............................................ 129
7.2. Narrow Parts of Convex Polygons of Restricted Hyperbolic Type in a Hyperbolic
Plane: General Results ............................................................. 132
7.3. Narrow Parts of Restricted Hyperbolic Convex Polygons in a Hyperbolic Plane:
Refined Results..........................................................................................140
7.4. Some General Applications ............................................................................155
b ^ reflective hyperbolic lattices over purely real algebraic num-
er holds (155). 7.4.2. Algebraic surfaces with an almost finite polyhedral Mori
cone (156)
7.5. Description of the Narrow Parts of Fundamental Polygons M of Reflection Subgroups
W C W(5) of Hyperbolic Type for rk S = 3. Application to Hyperbolically Reflective
Hyperbolic Lattices ................................................................. 156
7.5.1. Matrices B of narrow parts of type (All) (157). 7.5.2. Matrices B of narrow
parts of type (AIO) (157). 7.5.3. Matrices B of narrow parts of type (AII1) (158).
7.5.4. Matrices B of narrow parts of type (AIIO) (158). 7.5.5. Matrices B of narrow
parts of type (AIII) (158). 7.5.6. Matrices B of narrow parts of type (BI) (159).
7.5.7. Matrices B of narrow parts of type (BIIi) (159). 7.5.8. Matrices B of narrow
parts of type (BII2) (160). 7.5.9. Matrices B of narrow parts of type (Bill) (161).
7.5.10. The global estimate of invariants of hyperbolically reflective hyperbolic
lattices of rank 3 (161)
Chapter 8. CLASSIFICATION OF HYPERBOLICALLY REFLECTIVE HYPERBOLIC LAT-
TICES OF RANK THREE: PROOFS ................................................... 162
8.1. Proof of Basic Theorem 6.1 ......................................................... 162
8.2. Proof of Theorem 6.2 ............................................................... 168
8.3. Proof of Theorem 6.4 ............................................................... 169
Appendix II.l. TABLES 4-7........................................................................ 170
Table 4. The List of Hyperbolically Reflective Main Hyperbolic Lattices of Rank Three .. 170
Table 5. The List of Hyperbolically Reflective Odd Hyperbolic Lattices of Rank Three
with the Even Square-Free Determinant (i.e., Non-main) ....................... 190
Table 6. The List of Main Hyperbolic Lattices of Rank Three with the Determinant d
100000 and h = 2 .............................................................. 194
Table 7. Possible Types of Narrow Parts from Theorem 7.3.4 of Main Hyperbolic Lattices
of Rank 3 with h = 0 or 2...................................................... 201
Appendix II.2. PROGRAMS FOR THE GP/PARI CALCULATOR. HYPERBOLIC TYPE ......... 209
Program 13: fundall.gen ............................................................... 209
Program 14: fundalO.gen ............................................................... 210
Program 15: funda21.gen ............................................................... 211
Program 16: funda20.gen ............................................................... 212
Program 17: funda3.gen.................................................................213
Program 18: fundbl.gen ................................................................ 214
Program 19: fundb21.gen ............................................................... 215
Program 20: fundb22.gen ............................................................... 216
Program 21: fundb3.gen ................................................................ 217
Program 22: fundall.main .............................................................. 218
Program 23: fundalO.main .............................................................. 220
Program 24: funda21.main .............................................................. 222
Program 25: funda20.main .............................................................. 224
Program 26: funda3.main ............................................................... 226
Program 27: fundbl.main ............................................................... 228
Program 28: fundb21.main .............................................................. 231
Program 29: fundb22.main .............................................................. 233
Program 30: fundb3.main ............................................................... 235
Program 31: refl0.14 .................................................................... 237
REFERENCES ..................................................................................... 239
|
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| publisher | Interperiodica |
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| series | Matematičeskij Institut Imeni V. A. Steklova <Moskva>: Proceedings of the Steklov Institute of Mathematics |
| series2 | Matematičeskij Institut Imeni V. A. Steklova <Moskva>: Proceedings of the Steklov Institute of Mathematics |
| spellingShingle | Nikulin, Vjačeslav V. 1950- On the classification of hyperbolic root systems of rank three Matematičeskij Institut Imeni V. A. Steklova <Moskva>: Proceedings of the Steklov Institute of Mathematics |
| subject_GND | (DE-588)1071861417 |
| title | On the classification of hyperbolic root systems of rank three |
| title_auth | On the classification of hyperbolic root systems of rank three |
| title_exact_search | On the classification of hyperbolic root systems of rank three |
| title_full | On the classification of hyperbolic root systems of rank three V. V. Nikulin |
| title_fullStr | On the classification of hyperbolic root systems of rank three V. V. Nikulin |
| title_full_unstemmed | On the classification of hyperbolic root systems of rank three V. V. Nikulin |
| title_short | On the classification of hyperbolic root systems of rank three |
| title_sort | on the classification of hyperbolic root systems of rank three |
| topic_facet | Konferenzschrift |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=009171502&sequence=000001&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
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