Foundations of Lie theory, Lie transformation groups
著者
書誌事項
Foundations of Lie theory, Lie transformation groups
（Encyclopaedia of mathematical sciences / editor-in-chief, R.V. Gamkrelidze, v. 20 . Lie groups and Lie algebras ; 1）
Springer-Verlag, c1993
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Gruppy Li i algebry Li
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注記
Partial Contents: Foundations of Lie theory / A.L. Onishchik, E.B. Vinberg -- Lie transformation groups / V.V. Gorbatsevich, A.L. Onishchik
"Translation of original Russian, issued as v. 20 of the serial: Itogi nauki i tekhniki. Sovremennye problemy matematiki. Fundamentalʹnye napravlenii︠a︡. 1988" -- T.p. verso
Includes bibliographical references and indexes
内容説明・目次
内容説明
From the reviews: "..., the book must be of great help for a researcher who already has some idea of Lie theory, wants to employ it in his everyday research and/or teaching, and needs a source for customary reference on the subject. From my viewpoint, the volume is perfectly fit to serve as such a source, ... On the whole, it is quite a pleasure, after making yourself comfortable in that favourite office armchair of yours, just to keep the volume gently in your hands and browse it slowly and thoughtfully; and after all, what more on Earth can one expect of any book?" --The New Zealand Mathematical Society Newsletter
目次
I.Foundations of Lie Theory.- 1. Basic Notions.- 1. Lie Groups, Subgroups and Homomorphisms.- 1.1 Definition of a Lie Group.- 1.2 Lie Subgroups.- 1.3 Homomorphisms of Lie Groups.- 1.4 Linear Representations of Lie Groups.- 1.5 Local Lie Groups.- 2. Actions of Lie Groups.- 2.1 Definition of an Action.- 2.2 Orbits and Stabilizers.- 2.3 Images and Kernels of Homomorphisms.- 2.4 Orbits of Compact Lie Groups.- 3. Coset Manifolds and Quotients of Lie Groups.- 3.1 Coset Manifolds.- 3.2 Lie Quotient Groups.- 3.3 The Transitive Action Theorem and the Epimorphism Theorem.- 3.4 The Pre-image of a Lie Group Under a Homomorphism.- 3.5 Semidirect Products of Lie Groups.- 4. Connectedness and Simply-connectedness of Lie Groups.- 4.1 Connected Components of a Lie Group.- 4.2 Investigation of Connectedness of the Classical Lie Groups.- 4.3 Covering Homomorphisms.- 4.4 The Universal Covering Lie Group.- 4.5 Investigation of Simply-connectedness of the Classical Lie Groups.- 2. The Relation Between Lie Groups and Lie Algebras.- 1. The Lie Functor.- 1.1 The Tangent Algebra of a Lie Group.- 1.2 Vector Fields on a Lie Group.- 1.3 The Differential of a Homomorphism of Lie Groups.- 1.4 The Differential of an Action of a Lie Group.- 1.5 The Tangent Algebra of a Stabilizer.- 1.6 The Adjoint Representation.- 2. Integration of Homomorphisms of Lie Algebras.- 2.1 The Differential Equation of a Path in a Lie Group.- 2.2 The Uniqueness Theorem.- 2.3 Virtual Lie Subgroups.- 2.4 The Correspondence Between Lie Subgroups of a Lie Group and Subalgebras of Its Tangent Algebra.- 2.5 Deformations of Paths in Lie Groups.- 2.6 The Existence Theorem.- 2.7 Abelian Lie Groups.- 3. The Exponential Map.- 3.1 One-Parameter Subgroups.- 3.2 Definition and Basic Properties of the Exponential Map.- 3.3 The Differential of the Exponential Map.- 3.4 The Exponential Map in the Full Linear Group.- 3.5 Cartan's Theorem.- 3.6 The Subgroup of Fixed Points of an Automorphism of a Lie Group.- 4. Automorphisms and Derivations.- 4.1 The Group of Automorphisms.- 4.2 The Algebra of Derivations.- 4.3 The Tangent Algebra of a Semi-Direct Product of Lie Groups.- 5. The Commutator Subgroup and the Radical.- 5.1 The Commutator Subgroup.- 5.2 The Maltsev Closure.- 5.3 The Structure of Virtual Lie Subgroups.- 5.4 Mutual Commutator Subgroups.- 5.5 Solvable Lie Groups.- 5.6 The Radical.- 5.7 Nilpotent Lie Groups.- 3. The Universal Enveloping Algebra.- 1. The Simplest Properties of Universal Enveloping Algebras.- 1.1 Definition and Construction.- 1.2 The Poincare-Birkhoff-Witt Theorem.- 1.3 Symmetrization.- 1.4 The Center of the Universal Enveloping Algebra.- 1.5 The Skew-Field of Fractions of the Universal Enveloping Algebra.- 2. Bialgebras Associated with Lie Algebras and Lie Groups.- 2.1 Bialgebras.- 2.2 Right Invariant Differential Operators on a Lie Group.- 2.3 Bialgebras Associated with a Lie Group.- 3. The Campbell-Hausdorff Formula.- 3.1 Free Lie Algebras.- 3.2 The Campbell-Hausdorff Series.- 3.3 Convergence of the Campbell-Hausdorff Series.- 4. Generalizations of Lie Groups.- 1. Lie Groups over Complete Valued Fields.- 1.1 Valued Fields.- 1.2 Basic Definitions and Examples.- 1.3 Actions of Lie Groups.- 1.4 Standard Lie Groups over a Non-archimedean Field.- 1.5 Tangent Algebras of Lie Groups.- 2. Formal Groups.- 2.1 Definition and Simplest Properties.- 2.2 The Tangent Algebra of a Formal Group.- 2.3 The Bialgebra Associated with a Formal Group.- 3. Infinite-Dimensional Lie Groups.- 3.1 Banach Lie Groups.- 3.2 The Correspondence Between Banach Lie Groups and Banach Lie Algebras.- 3.3 Actions of Banach Lie Groups on Finite-Dimensional Manifolds.- 3.4 Lie-Frechet Groups.- 3.5 ILB- and ILH-Lie Groups.- 4. Lie Groups and Topological Groups.- 4.1 Continuous Homomorphisms of Lie Groups.- 4.2 Hilbert's 5-th Problem.- 5. Analytic Loops.- 5.1 Basic Definitions and Examples.- 5.2 The Tangent Algebra of an Analytic Loop.- 5.3 The Tangent Algebra of a Diassociative Loop.- 5.4 The Tangent Algebra of a Bol Loop.- References.- II. Lie Transformation Groups.- 1. Lie Group Actions on Manifolds.- 1. Introductory Concepts.- 1.1 Basic Definitions.- 1.2 Some Examples and Special Cases.- 1.3 Local Actions.- 1.4 Orbits and Stabilizers.- 1.5 Representation in the Space of Functions.- 2. Infinitesimal Study of Actions.- 2.1 Flows and Vector Fields.- 2.2 Infinitesimal Description of Actions and Morphisms.- 2.3 Existence Theorems.- 2.4 Groups of Automorphisms of Certain Geometric Structures.- 3. Fibre Bundles.- 3.1 Fibre Bundles with a Structure Group.- 3.2 Examples of Fibre Bundles.- 3.3 G-bundles.- 3.4 Induced Bundles and the Classification Theorem.- 2. Transitive Actions.- 1. Group Models.- 1.1 Definitions and Examples.- 1.2 Basic Problems.- 1.3 The Group of Automorphisms.- 1.4 Primitive Actions.- 2. Some Facts Concerning Topology of Homogeneous Spaces.- 2.1 Covering Spaces.- 2.2 Real Cohomology of Lie Groups.- 2.3 Subgroups with Maximal Exponent in Simple Lie Groups.- 2.4 Some Homotopy Invariants of Homogeneous Spaces.- 3. Homogeneous Bundles.- 3.1 Invariant Sections and Classification of Homogeneous Bundles.- 3.2 Homogeneous Vector Bundles. The Frobenius Duality.- 3.3 The Linear Isotropy Representation and Invariant Vector Fields.- 3.4 Invariant. A-structures.- 3.5 Invariant Integration.- 3.6 Karpelevich-Mostow Bundles.- 4. Inclusions Among Transitive Actions.- 4.1 Reductions of Transitive Actions and Factorization of Groups.- 4.2 The Natural Enlargement of an Action.- 4.3 Some Inclusions Among Transitive Actions on Spheres.- 4.4 Factorizations of Lie Groups and Lie Algebras.- 4.5 Factorizations of Compact Lie Groups.- 4.6 Compact Enlargements of Transitive Actions of Simple Lie Groups.- 4.7 Groups of Isometries of Riemannian Homogeneous Spaces of Simple Compact Lie Groups.- 4.8 Groups of Automorphisms of Simply Connected Homogeneous Compact Complex Manifolds.- 3. Actions of Compact Lie Groups.- 1. The General Theory of Compact Lie Transformation Groups.- 1.1 Proper Actions.- 1.2 Existence of Slices.- 1.3 Two Fiberings of an Equi-orbital G-space.- 1.4 Principal Orbits.- 1.5 Orbit Structure.- 1.6 Linearization of Actions.- 1.7 Lifting of Actions.- 2. Invariants and Almost-Invariants.- 2.1 Applications of Invariant Integration.- 2.2 Finiteness Theorems for Invariants.- 2.3 Finiteness Theorems for Almost Invariants.- 3. Applications to Homogeneous Spaces of Reductive Groups.- 3.1 Complexification of Homogeneous Spaces.- 3.2 Factorization of Reductive Algebraic Groups and Lie Algebras.- 4. Homogeneous Spaces of Nilpotent and Solvable Groups.- 1. Nilmanifolds.- 1.1 Examples of Nilmanifolds.- 1.2 Topology of Arbitrary Nilmanifolds.- 1.3 Structure of Compact Nilmanifolds.- 1.4 Compact Nilmanifolds as Towers of Principal Bundles with Fibre T1.- 2. Solvmanifolds.- 2.1 Examples of Solvmanifolds.- 2.2 Solvmanifolds and Vector Bundles.- 2.3 Compact Solvmanifolds (The Structure Theorem).- 2.4 The Fundamental Group of a Solvmanifold.- 2.5 The Tangent Bundle of a Compact Solvmanifold.- 2.6 Transitive Actions of Lie Groups on Compact Solvmanifolds.- 2.7 The Case of Discrete Stabilizers.- 2.8 Homogeneous Spaces of Solvable Lie Groups of Type (I).- 2.9 Complex Compact Solvmanifolds.- 5. Compact Homogeneous Spaces.- 1. Uniform Subgroups.- 1.1 Algebraic Uniform Subgroups.- 1.2 Tits Bundles.- 1.3 Uniform Subgroups of Semi-simple Lie Groups.- 1.4 Connected Uniform Subgroups.- 1.5 Reductions of Transitive Actions of Reductive Lie Groups.- 2. Transitive Actions on Compact Homogeneous Spaces with Finite Fundamental Groups.- 2.1 Three Lemmas on Transitive Actions.- 2.2 Radical Enlargements.- 2.3 A Sufficient Condition for the Radical to be Abelian.- 2.4 Passage from Compact Groups to Non-Compact Semi-simple Groups.- 2.5 Compact Homogeneous Spaces of Rank 1.- 2.6 Transitive Actions of Non-Compact Lie Groups on Spheres.- 2.7 Existence of Maximal and Largest Enlargements.- 3. The Natural Bundle.- 3.1 Orbits of the Action of a Maximal Compact Subgroup.- 3.2 Construction of the Natural Bundle and Its Properties.- 3.3 Some Examples of Natural Bundles.- 3.4 On the Uniqueness of the Natural Bundle.- 3.5 The Case of Low Dimension of Fibre and Basis.- 4. The Structure Bundle.- 4.1 Regular Transitive Actions of Lie Groups.- 4.2 The Structure of the Base of the Natural Bundle.- 4.3 Some Examples of Structure Bundles.- 5. The Fundamental Group.- 5.1 On the Concept of Commensurability of Groups.- 5.2 Embedding of the Fundamental Group in a Lie Group.- 5.3 Solvable and Semi-simple Components.- 5.4 Cohomological Dimension.- 5.5 The Euler Characteristic.- 5.6 The Number of Ends.- 6. Some Classes of Compact Homogeneous Spaces.- 6.1 Three Components of a Compact Homogeneous Space and the Case when Two of them Are Trivial.- 6.2 The Case of One Trivial Component.- 7. Aspherical Compact Homogeneous Spaces.- 7.1 Group Models of Aspherical Compact Homogeneous Spaces.- 7.2 On the Fundamental Group.- 8. Semi-simple Compact Homogeneous Spaces.- 8.1 Transitivity of a Semi-simple Subgroup.- 8.2 The Fundamental Group.- 8.3 On the Fibre of the Natural Bundle.- 9. Solvable Compact Homogeneous Spaces.- 9.1 Properties of the Natural Bundle.- 9.2 Elementary Solvable Homogeneous Spaces.- 10. Compact Homogeneous Spaces with Discrete Stabilizers.- 6. Actions of Lie Groups on Low-dimensional Manifolds.- 1. Classification of Local Actions.- 1.1 Notes on Local Actions.- 1.2 Classification of Local Actions of Lie Groups on ?1, ?1.- 1.3 Classification of Local Actions of Lie Groups on ?2 and ?2.- 2. Homogeneous Spaces of Dimension ?3.- 2.1 One-dimensional Homogeneous Spaces.- 2.2 Two-dimensional Homogeneous Spaces (Homogeneous Surfaces).- 2.3 Three-dimensional Manifolds.- 3. Compact Homogeneous Manifolds of Low Dimension.- 3.1 On Four-dimensional Compact Homogeneous Manifolds.- 3.2 Compact Homogeneous Manifolds of Dimension ?6.- 3.3 On Compact Homogeneous Manifolds of Dimension ?7.- References.
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