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Lie algebras and algebraic groups:

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Main Authors:	Tauvel, Patrice (Author), Yu, Rupert W. T. (Author)
Format:	Book
Language:	English
Published:	Berlin [u. a.] Springer 2005
Series:	Springer monographs in mathematics
Subjects:	Algebraische Gruppe Lie-Algebra Lie algebras Linear algebraic groups
Online Access:	Inhaltsverzeichnis
Item Description:	Auch als Internetausgabe
Physical Description:	XVI, 653 S.
ISBN:	9783540241706 3540241701

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Record in the Search Index

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adam_text	Contents 1 Results on topological spaces .............................. 1 1.1 Irreducible sets and spaces ............................... 1 1.2 Dimension ............................................. 4 1.3 Noetherian spaces ....................................... 5 1.4 Constructible sets ....................................... 6 1.5 Gluing topological spaces ................................ 8 2 Rings and modules ........................................ 11 2.1 Ideals ................................................. 11 2.2 Prime and maximal ideals ................................ 12 2.3 Rings of fractions and localization ......................... 13 2.4 Localizations of modules ................................. 17 2.5 Radical of an ideal ...................................... 18 2.6 Local rings ............................................. 19 2.7 Noetherian rings and modules ............................ 21 2.8 Derivations ............................................ 24 2.9 Module of differentials ................................... 25 3 Integral extensions ........................................ 31 3.1 Integral dependence ..................................... 31 3.2 Integrally closed domains ................................ 33 3.3 Extensions of prime ideals ............................... 35 4 Factorial rings ............................................. 39 4.1 Generalities ............................................ 39 4.2 Unique factorization ..................................... 41 4.3 Principal ideal domains and Euclidean domains ............. 43 4.4 Polynomials and factorial rings ........................... 45 4.5 Symmetric polynomials .................................. 48 4.6 Resultant and discriminant ............................... 50 χ Contents 5 Field extensions ........................................... 55 5.1 Extensions ............................................. 55 5.2 Algebraic and transcendental elements ..................... 56 5.3 Algebraic extensions .................................... 56 5.4 Transcendence basis ..................................... 58 5.5 Norm and trace ......................................... 60 5.6 Theorem of the primitive element ......................... 62 5.7 Going Down Theorem ................................... 64 5.8 Fields and derivations ................................... 67 5.9 Conductor ............................................. 70 6 Finitely generated algebras ................................ 75 6.1 Dimension ............................................. 75 6.2 Noether s Normalization Theorem ......................... 76 6.3 Krull s Principal Ideal Theorem ........................... 81 6.4 Maximal ideals ......................................... 82 6.5 Zariski topology ........................................ 84 7 Gradings and nitrations ................................... 87 7.1 Graded rings and graded modules ......................... 87 7.2 Graded submodules ..................................... 88 7.3 Applications ........................................... 90 7.4 Filtrations ............................................. 91 7.5 Grading associated to a filtration ......................... 92 8 Inductive limits ............................................ 95 8.1 Generalities ............................................ 95 8.2 Inductive systems of maps ............................... 96 8.3 Inductive systems of magmas, groups and rings ............. 98 8.4 An example ............................................100 8.5 Inductive systems of algebras .............................100 9 Sheaves of functions .......................................103 9.1 Sheaves ................................................103 9.2 Morphisms .............................................104 9.3 Sheaf associated to a presheaf ............................106 9.4 Gluing ................................................109 9.5 Ringed space ...........................................110 10 Jordan decomposition and some basic results on groups ... 113 10.1 Jordan decomposition ...................................113 10.2 Generalities on groups ...................................117 10.3 Commutators ..........................................118 10.4 Solvable groups .........................................120 10.5 Nilpotent groups ........................................121 Contents XI 10.6 Group actions ..........................................122 10.7 Generalities on representations ...........................123 10.8 Examples ..............................................126 11 Algebraic sets .............................................131 11.1 Affine algebraic sets .....................................131 11.2 Zariski topology ........................................132 11.3 Regular functions .......................................133 11.4 Morphisms .............................................134 11.5 Examples of morphisms ..................................136 11.6 Abstract algebraic sets ..................................138 11.7 Principal open subsets ...................................140 11.8 Products of algebraic sets ................................142 12 Prevarieties and varieties ..................................147 12.1 Structure sheaf .........................................147 12.2 Algebraic prevarieties ...................................149 12.3 Morphisms of prevarieties ................................151 12.4 Products of prevarieties ..................................152 12.5 Algebraic varieties ......................................155 12.6 Gluing ................................................158 12.7 Rational functions ......................................159 12.8 Local rings of a variety ..................................162 13 Projective varieties ........................................167 13.1 Projective spaces .......................................167 13.2 Projective spaces and varieties ............................168 13.3 Cones and projective varieties ............................171 13.4 Complete varieties ......................................176 13.5 Products ...............................................178 13.6 Grassmannian variety ...................................180 14 Dimension .................................................183 14.1 Dimension of varieties ...................................183 14.2 Dimension and the number of equations ...................185 14.3 System of parameters ...................................187 14.4 Counterexamples .......................................190 15 Morphisms and dimension .................................191 15.1 Criterion of affmeness ...................................191 15.2 Affine morphisms .......................................193 15.3 Finite morphisms .......................................194 15.4 Factorization and applications ............................197 15.5 Dimension of fibres of a morphism ........................199 15.6 An example ............................................203 XII Contents 16 Tangent spaces ............................................205 16.1 A first approach ........................................205 16.2 Zariski tangent space ....................................207 16.3 Differential of a morphism ...............................209 16.4 Some lemmas ..........................................213 16.5 Smooth points ..........................................215 17 Normal varieties ...........................................219 17.1 Normal varieties ........................................219 17.2 Normalization ..........................................221 17.3 Products of normal varieties ..............................223 17.4 Properties of normal varieties ............................225 18 Root systems ..............................................233 18.1 Reflections ............................................ . 233 18.2 Root systems ...........................................235 18.3 Root systems and bilinear forms ..........................238 18.4 Passage to the field of real numbers .......................239 18.5 Relations between two roots ..............................240 18.6 Examples of root systems ................................243 18.7 Base of a root system ...................................244 18.8 Weyl chambers .........................................247 18.9 Highest root ...........................................250 18.10 Closed subsets of roots ..................................250 18.11 Weights ...............................................253 18.12 Graphs ................................................255 18.13 Dynkin diagrams .......................................256 18.14 Classification of root systems .............................259 19 Lie algebras ...............................................277 19.1 Generalities on Lie algebras ..............................277 19.2 Representations .........................................279 19.3 Nilpotent Lie algebras ...................................282 19.4 Solvable Lie algebras ....................................286 19.5 Radical and the largest nilpotent ideal .....................289 19.6 Nilpotent radical ........................................291 19.7 Regular linear forms .....................................292 19.8 Cartari subalgebras .....................................294 20 Semisimple and reductive Lie algebras .....................299 20.1 Semisimple Lie algebras .................................299 20.2 Examples ..............................................301 20.3 Semisimplicity of representations ..........................302 20.4 Semisimple and nilpotent elements ........................305 20.5 Reductive Lie algebras ...................................307 Contents XIII 20.6 Results on the structure of semisimple Lie algebras ..........310 20.7 Subalgebras of semisimple Lie algebras ....................313 20.8 Parabolic subalgebras ...................................316 21 Algebraic groups ..........................................319 21.1 Generalities ............................................319 21.2 Subgroups and morphisms ...............................321 21.3 Connectedness ..........................................322 21.4 Actions of an algebraic group .............................325 21.5 Modules ...............................................326 21.6 Group closure ..........................................327 22 Affine algebraic groups ....................................331 22.1 Translations of functions .................................331 22.2 Jordan decomposition ...................................333 22.3 Unipotent groups .......................................335 22.4 Characters and weights ..................................338 22.5 Tori and diagonalizable groups ...........................340 22.6 Groups of dimension one .................................345 23 Lie algebra of an algebraic group ..........................347 23.1 An associative algebra ...................................347 23.2 Lie algebras ............................................348 23.3 Examples ..............................................352 23.4 Computing differentials ..................................354 23.5 Adjoint representation ...................................359 23.6 Jordan decomposition ...................................362 24 Correspondence between groups and Lie algebras ..........365 24.1 Notations ..............................................365 24.2 An algebraic subgroup ...................................365 24.3 Invariants ..............................................368 24.4 Functorial properties ....................................372 24.5 Algebraic Lie subalgebras ................................375 24.6 A particular case .......................................380 24.7 Examples ..............................................383 24.8 Algebraic adjoint group ..................................383 25 Homogeneous spaces and quotients ........................387 25.1 Homogeneous spaces ....................................387 25.2 Some remarks ..........................................389 25.3 Geometric quotients .....................................391 25.4 Quotient by a subgroup ..................................393 25.5 The case of finite groups .................................397 XIV Contents 26 Solvable groups ............................................401 26.1 Conjugacy classes .......................................401 26.2 Actions of diagonalizable groups ..........................405 26.3 Fixed points ...........................................406 26.4 Properties of solvable groups .............................407 26.5 Structure of solvable groups ..............................409 27 Reductive groups ..........................................413 27.1 Radical and unipotent radical ............................413 27.2 Semisimple and reductive groups ..........................415 27.3 Representations .........................................416 27.4 Fimteness properties ....................................420 27.5 Algebraic quotients .....................................422 27.6 Characters .............................................424 28 Borei subgroups, parabolic subgroups, Cartari subgroups . . 429 28.1 Borei subgroups ........................................429 28.2 Theorems of density .....................................432 28.3 Centralizers and tori ....................................434 28.4 Properties of parabolic subgroups .........................435 28.5 Cartan subgroups .......................................437 29 Cartan subalgebras, Borei subalgebras and parabolic subalgebras ................................................441 29.1 Generalities ............................................441 29.2 Cartari subalgebras .....................................443 29.3 Applications to semisimple Lie algebras ....................446 29.4 Borei subalgebras .......................................447 29.5 Properties of parabolic subalgebras ........................450 29.6 More on reductive Lie algebras ...........................453 29.7 Other applications ......................................454 29.8 Maximal subalgebras ....................................456 30 Representations of semisimple Lie algebras ................459 30.1 Enveloping algebra ......................................459 30.2 Weights and primitive elements ...........................461 30.3 Finite-dimensional modules ..............................463 30.4 Verma modules .........................................464 30.5 Results on existence and uniqueness .......................467 30.6 A property of the Weyl group ............................469 31 Symmetric invariants ......................................471 31.1 Invariants of finite groups ................................471 31.2 Invariant polynomial functions ............................475 31.3 A free module ..........................................478 Contents XV 32 S-triples ...................................................481 32.1 Jacobson-Morosov Theorem ..............................481 32.2 Some lemmas ..........................................484 32.3 Conjugation of S-triples .................................487 32.4 Characteristic ..........................................488 32.5 Regular and principal elements ...........................489 33 Polarizations .......................................·......493 33.1 Definition of polarizations ................................493 33.2 Polarizations in the semisimple case .......................494 33.3 A non-polarizable element ...............................497 33.4 Polarizable elements .....................................499 33.5 Richardson s Theorem ...................................502 34 Results on orbits ..........................................507 34.1 Notations ..............................................507 34.2 Some lemmas ..........................................508 34.3 Generalities on orbits ....................................509 34.4 Minimal nilpotent orbit .....................·............511 34.5 Subregular nilpotent orbit ...............................513 34.6 Dimension of nilpotent orbits .............................517 34.7 Prehomogeneous spaces of parabolic type ..................518 35 Centralizers ...............................................521 35.1 Distinguished elements ..................................521 35.2 Distinguished parabolic subalgebras ,......................523 35.3 Double centralizers ......................................525 35.4 Normalizes ............................................528 35.5 A semisimple Lie subalgebra .............................530 35.6 Centralizers and regular elements .........................533 36 er-root systems .............................................537 36.1 Definition ..............................................537 36.2 Restricted root systems ..................................539 36.3 Restriction of a root .....................................544 37 Symmetric Lie algebras ....................................549 37.1 Primary subspaces ......................................549 37.2 Definition of symmetric Lie algebras .......................553 37.3 Natural subalgebras .....................................554 37.4 Cartan subspaces .......................................555 37.5 The case of reductive Lie algebras .........................557 37.6 Linear forms ...........................................559 XVI Contents 38 Semisimple symmetric Lie algebras ........................561 38.1 Notations ..............................................561 38.2 Iwasawa decomposition ..................................562 38.3 Coroots ................................................565 38.4 Centralizers ............................................568 38.5 S-triples ...............................................570 38.6 Orbits .................................................573 38.7 Symmetric invariants ....................................579 38.8 Double centralizers ......................................584 38.9 Normalizers ............................................588 38.10 Distinguished elements ..................................589 39 Sheets of Lie algebras ......................................593 39.1 Jordan classes ..........................................593 39.2 Topology of Jordan classes ...............................596 39.3 Sheets .................................................601 39.4 Dixmier sheets .........................................603 39.5 Jordan classes in the symmetric case ......................605 39.6 Sheets in the symmetric case .............................608 40 Index and linear forms .....................................611 40.1 Stable linear forms ......................................611 40.2 Index of a representation ................................615 40.3 Some useful inequalities .................................616 40.4 Index and semi-direct products ...........................618 40.5 Heisenberg algebras in semisimple Lie algebras .............621 40.6 Index of Lie subalgebras of Borei subalgebras ...............625 40.7 Seaweed Lie algebras ....................................629 40.8 An upper bound for the index ............................630 40.9 Cases where the bound is exact ...........................635 40.10 On the index of parabolic subalgebras .....................638 References .....................................................641 List of notations ...............................................645 Index ..........................................................647
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spelling	Tauvel, Patrice Verfasser aut Lie algebras and algebraic groups Patrice Tauvel ; Rupert W. T. Yu Berlin [u. a.] Springer 2005 XVI, 653 S. txt rdacontent n rdamedia nc rdacarrier Springer monographs in mathematics Auch als Internetausgabe Algebraische Gruppe Lie-Algebra Lie algebras Linear algebraic groups Lie-Algebra (DE-588)4130355-6 gnd rswk-swf Algebraische Gruppe (DE-588)4001164-1 gnd rswk-swf Lie-Algebra (DE-588)4130355-6 s DE-604 Algebraische Gruppe (DE-588)4001164-1 s Yu, Rupert W. T. Verfasser aut Digitalisierung UB Augsburg application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=013324371&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis
spellingShingle	Tauvel, Patrice Yu, Rupert W. T. Lie algebras and algebraic groups Algebraische Gruppe Lie-Algebra Lie algebras Linear algebraic groups Lie-Algebra (DE-588)4130355-6 gnd Algebraische Gruppe (DE-588)4001164-1 gnd
subject_GND	(DE-588)4130355-6 (DE-588)4001164-1
title	Lie algebras and algebraic groups
title_auth	Lie algebras and algebraic groups
title_exact_search	Lie algebras and algebraic groups
title_full	Lie algebras and algebraic groups Patrice Tauvel ; Rupert W. T. Yu
title_fullStr	Lie algebras and algebraic groups Patrice Tauvel ; Rupert W. T. Yu
title_full_unstemmed	Lie algebras and algebraic groups Patrice Tauvel ; Rupert W. T. Yu
title_short	Lie algebras and algebraic groups
title_sort	lie algebras and algebraic groups
topic	Algebraische Gruppe Lie-Algebra Lie algebras Linear algebraic groups Lie-Algebra (DE-588)4130355-6 gnd Algebraische Gruppe (DE-588)4001164-1 gnd
topic_facet	Algebraische Gruppe Lie-Algebra Lie algebras Linear algebraic groups
url	http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=013324371&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA
work_keys_str_mv	AT tauvelpatrice liealgebrasandalgebraicgroups AT yurupertwt liealgebrasandalgebraicgroups

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