Topics in noncommutative algebra: the theorem of Campbell, Baker, Hausdorff and Dynkin
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| Main Authors: | , |
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| Format: | Book |
| Language: | English |
| Published: |
Berlin [u.a.]
Springer
2012
|
| Series: | Lecture Notes in Mathematics
2034 |
| Subjects: | |
| Online Access: | Inhaltstext Inhaltsverzeichnis |
| Physical Description: | XXII, 539 S. graph. Darst. |
| ISBN: | 3642225969 9783642225963 |
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MARC
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IMAGE 1
CONTENTS
1 HISTORICAL OVERVIEW 1
1.1 THE EARLY PROOFS OF THE CBHD THEOREM 1
1.1.1 THE ORIGIN OF THE PROBLEM 2
1.1.2 SCHUR, POINCARE AND PASCAL 9
1.1.3 CAMPBELL, BAKER, HAUSDORFF, DYNKIN 19
1.2 THE "MODERN ERA" OF THE CBHD THEOREM 37
1.3 THE "NAME OF THE GAME" 44
PART I ALGEBRAIC PROOFS OF THE THEOREM OF CAMPBELL, BAKER, HAUSDORFF AND
DYNKIN
2 BACKGROUND ALGEBRA 49
2.1 FREE VECTOR SPACES, ALGEBRAS AND TENSOR PRODUCTS 49 2.1.1 VECTOR
SPACES AND FREE VECTOR SPACES 49
2.1.2 MAGMAS, ALGEBRAS AND (UNITAL) ASSOCIATIVE ALGEBRAS 56
2.1.3 TENSOR PRODUCT AND TENSOR ALGEBRA 72
2.2 FREE LIE ALGEBRAS 87
2.3 COMPLETIONS OF GRADED TOPOLOGICAL ALGEBRAS 93
2.3.1 TOPOLOGY ON SOME CLASSES OF ALGEBRAS 94
2.3.2 COMPLETIONS OF GRADED TOPOLOGICAL ALGEBRAS 98 2.3.3 FORMAL POWER
SERIES 101
2.3.4 SOME MORE NOTATION ON FORMAL POWER SERIES 106 2.4 THE UNIVERSAL
ENVELOPING ALGEBRA 108
3 THE MAIN PROOF OF THE CBHD THEOREM 115
3.1 EXPONENTIAL AND LOGARITHM 117
3.1.1 EXPONENTIALS AND LOGARITHMS 119
3.1.2 THE STATEMENT OF OURMAIN CBHD THEOREM 124 3.1.3 THE OPERATION* ON
$+(V) 126
3.1.4 THE OPERATION O ON &+(V) 128
BIBLIOGRAFISCHE INFORMATIONEN HTTP://D-NB.INFO/1012892735
DIGITALISIERT DURCH
IMAGE 2
XX CONTENTS
3.2 THE CAMPBELL, BAKER, HAUSDORFF THEOREM 132
3.2.1 FRIEDRICHS'S CHARACTERIZATION OF LIE ELEMENTS 133 3.2.2 THE
CAMPBELL, BAKER, HAUSDORFF THEOREM 139 3.2.3 THE HAUSDORFF GROUP 142
3.3 DYNKIN'S FORMULA 145
3.3.1 THE LEMMA OF DYNKIN, SPECHT, WEVER 145
3.3.2 DYNKIN'S FORMULA 151
3.3.3 THE FINAL PROOF OF THE CBHD THEOREM 154
3.3.4 SOME "FINITE" IDENTITIES ARISING FROM THE EQUALITY BETWEEN * AND O
156
3.4 RESUME: THE "SPINE" OF THE PROOF OF THE CBHD THEOREM. 158 3.5 A
FEW SUMMANDS OF THE DYNKIN SERIES 159
3.6 FURTHER READING: HOPF ALGEBRAS 162
4 SOME "SHORT" PROOFS OF THE CBHD THEOREM 173
4.1 STATEMENT OF THE CBHD THEOREM FOR FORMAL POWER SERIES IN TWO
INDETERMINATES 178
4.2 EICHLER'S PROOF 187
4.2.1 EICHLER'S INDUCTIVE ARGUMENT 189
4.3 DJOKOVIC'S PROOF 199
4.3.1 POLYNOMIALS AND SERIES IN T OVER A UA ALGEBRA 199 4.3.2 BACKGROUND
OF DJOKOVIC'S PROOF 205
4.3.3 DJOKOVIC'S ARGUMENT 210
4.4 THE "SPINE" OF THE PROOF 216
4.4.1 YET ANOTHER PROOF WITH FORMAL POWER SERIES 220 4.5 VARADARAJAN'S
PROOF 223
4.5.1 A RECURSION FORMULA FOR THE CBHD SERIES 227 4.5.2 ANOTHER
RECURSION FORMULA 229
4.6 REUTENAUER'S PROOF 231
4.7 CARRIER'S PROOF 253
4.7.1 SOME IMPORTANT MAPS 254
4.7.2 A NEW CHARACTERIZATION OF LIE ELEMENTS 258
5 CONVERGENCE OF THE CBHD SERIES AND ASSOCIATIVITY OF THE CBHD OPERATION
265
5.1 "FINITE" IDENTITIES OBTAINED FROM THE CBHD THEOREM 268 5.2
CONVERGENCE OF THE CBHD SERIES 277
5.2.1 THE CASE OF FINITE DIMENSIONAL LIE ALGEBRAS 278 5.2.2 THE CASE OF
BANACH-LIE ALGEBRAS 292
5.2.3 AN IMPROVED DOMAIN OF CONVERGENCE 301
5.3 ASSOCIATIVITY OF THE CBHD OPERATION 305
5.3.1 "FINITE" IDENTITIES FROM THE ASSOCIATIVITY OF O 306
5.3.2 ASSOCIATIVITY FOR BANACH-LIE ALGEBRAS 312
IMAGE 3
CONTENTS XXI
5.4 NILPOTENT LIE ALGEBRAS AND THE THIRD THEOREM OF LIE 320 5.4.1
ASSOCIATIVITY FOR NILPOTENT LIE ALGEBRAS 320 5.4.2 THE GLOBAL THIRD
THEOREM OF LIE FOR NILPOTENT LIE ALGEBRAS 328
5.5 THE CBHD OPERATION AND SERIES IN BANACH ALGEBRAS 337 5.5.1 AN
ALTERNATIVE APPROACH USING ANALYTIC FUNCTIONS 347
5.6 AN EXAMPLE OF NON-CONVERGENCE OF THE CBHD SERIES 354 5.7 FURTHER
REFERENCES 359
6 RELATIONSHIP BETWEEN THE CBHD THEOREM, THE PBW THEOREM AND THE FREE
LIE ALGEBRAS 371
6.1 PROVING PBW BY MEANS OF CBHD 375
6.1.1 SOME PRELIMINARIES 375
6.1.2 CARTIER'S PROOF OF PBW VIA CBHD 383
PART II PROOFS OF THE ALGEBRAIC PREREQUISITES
7 PROOFS OF THE ALGEBRAIC PREREQUISITES 393
7.1 PROOFS OF SECT. 2.1.1 393
7.2 PROOFS OF SECT. 2.1.2 396
7.3 PROOFS OF SECT. 2.1.3 396
7.4 PROOFS OF SECT. 2.3.1 407
7.5 PROOFS OF SECT. 2.3.2 417
7.6 PROOFS OF SECT. 2.3.3 428
7.7 PROOFS OF SECT. 2.4 435
7.8 MISCELLANEA OF PROOFS 445
8 CONSTRUCTION OF FREE LIE ALGEBRAS 459
8.1 CONSTRUCTION OF FREE LIE ALGEBRAS CONTINUED 459 8.1.1 FREE LIE
ALGEBRAS OVER A SET 463
8.2 FREE NILPOTENT LIE ALGEBRA GENERATED BY A SET 469
9 FORMAL POWER SERIES IN ONE INDETERMINATE 479
9.1 OPERATIONS ON FORMAL POWER SERIES IN ONE INDETERMINATE 480
9.1.1 THE CAUCHY PRODUCT OF FORMAL POWER SERIES 480 9.1.2 SUBSTITUTION
OF FORMAL POWER SERIES 481
9.1.3 THE DERIVATION OPERATOR ON FORMAL POWER SERIES 486
9.1.4 THE RELATION BETWEEN THE EXP AND THE LOG SERIES . 488 9.2
BERNOULLI NUMBERS 494
IMAGE 4
XXII CONTENTS
10 SYMMETRIC ALGEBRA 501
10.1 THE SYMMETRIC ALGEBRA AND THE SYMMETRIC TENSOR SPACE. 501 10.1.1
BASIS FOR THE SYMMETRIC ALGEBRA 512
10.2 PROOFS OF SECT. 10.1 514
A LIST OF THE BASIC NOTATION 523
REFERENCE 529
INDEX 537 |
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| dewey-full | 512.48 |
| dewey-hundreds | 500 - Natural sciences and mathematics |
| dewey-ones | 512 - Algebra |
| dewey-raw | 512.48 |
| dewey-search | 512.48 |
| dewey-sort | 3512.48 |
| dewey-tens | 510 - Mathematics |
| discipline | Mathematik |
| format | Book |
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| physical | XXII, 539 S. graph. Darst. |
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| spelling | Bonfiglioli, Andrea 1974- Verfasser (DE-588)101850284X aut Topics in noncommutative algebra the theorem of Campbell, Baker, Hausdorff and Dynkin Andrea Bonfiglioli ; Roberta Fulci Berlin [u.a.] Springer 2012 XXII, 539 S. graph. Darst. txt rdacontent n rdamedia nc rdacarrier Lecture notes in mathematics 2034 Nichtkommutative Algebra (DE-588)4304013-5 gnd rswk-swf Nichtkommutative Algebra (DE-588)4304013-5 s DE-604 Fulci, Roberta Verfasser (DE-588)1018502890 aut Erscheint auch als Online-Ausgabe 978-3-642-22597-0 Lecture Notes in Mathematics 2034 (DE-604)BV000676446 2034 X:MVB text/html http://deposit.dnb.de/cgi-bin/dokserv?id=3843087&prov=M&dok_var=1&dok_ext=htm Inhaltstext DNB Datenaustausch application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=024534133&sequence=000001&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Bonfiglioli, Andrea 1974- Fulci, Roberta Topics in noncommutative algebra the theorem of Campbell, Baker, Hausdorff and Dynkin Lecture Notes in Mathematics Nichtkommutative Algebra (DE-588)4304013-5 gnd |
| subject_GND | (DE-588)4304013-5 |
| title | Topics in noncommutative algebra the theorem of Campbell, Baker, Hausdorff and Dynkin |
| title_auth | Topics in noncommutative algebra the theorem of Campbell, Baker, Hausdorff and Dynkin |
| title_exact_search | Topics in noncommutative algebra the theorem of Campbell, Baker, Hausdorff and Dynkin |
| title_full | Topics in noncommutative algebra the theorem of Campbell, Baker, Hausdorff and Dynkin Andrea Bonfiglioli ; Roberta Fulci |
| title_fullStr | Topics in noncommutative algebra the theorem of Campbell, Baker, Hausdorff and Dynkin Andrea Bonfiglioli ; Roberta Fulci |
| title_full_unstemmed | Topics in noncommutative algebra the theorem of Campbell, Baker, Hausdorff and Dynkin Andrea Bonfiglioli ; Roberta Fulci |
| title_short | Topics in noncommutative algebra |
| title_sort | topics in noncommutative algebra the theorem of campbell baker hausdorff and dynkin |
| title_sub | the theorem of Campbell, Baker, Hausdorff and Dynkin |
| topic | Nichtkommutative Algebra (DE-588)4304013-5 gnd |
| topic_facet | Nichtkommutative Algebra |
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