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· 분류 : 외국도서 > 과학/수학/생태 > 수학 > 대수학 > 선형대수학
· ISBN : 9783540612223
· 쪽수 : 238쪽
· 출판일 : 1996-12-18
책 소개
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
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
"... Both parts are very nicely written and can be strongly recommended."
European Mathematical Society
New feature
From the reviews:"This volume consists of two parts. ... Part I is devoted to a systematic development of the theory of Lie groups. The Lie algebras are studied only in connection with Lie groups, i.e. a systematic study of the Lie algebras is included here. Neither the structural theory of the Lie groups and Lie algebras nor a systematic study of the topology of Lie groups form the subject of this volume. On the other hand, Part I contains a very interesting chapter on generalizations of Lie groups including very recent results. We find here Lie groups over non-archimedian fields, formal groups, infinite dimensional Lie groups and also analytic loops. Part II deals on an advanced level with actions of Lie groups on manifolds and includes subjec ts like Lie groups actions on manifolds, transitive actions, actions of compact Lie groups on low-dimensional manifolds. Though the authors state that the geometry and topology of Lie groups is almost entirely beyond the scope of this survey, one can learn a lot in these directions. Both parts are very nicely written and can be strongly recommended."
European Mathematical Society Newsletter, 1993 "... 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... This is a hand- rather than a textbook. ... 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, 1994
목차
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.
