Linear Algebraic Groups by James E. Humphreys (PDF)

Description

I. Algebraic Geometry.- 0. Some Commutative Algebra.- 1. Affine and Projective Varieties.- 1.1 Ideals and Affine Varieties.- 1.2 Zariski Topology on Affine Space.- 1.3 Irreducible Components.- 1.4 Products of Affine Varieties.- 1.5 Affine Algebras and Morphisms.- 1.6 Projective Varieties.- 1.7 Products of Projective Varieties.- 1.8 Flag Varieties.- 2. Varieties.- 2.1 Local Rings.- 2.2 Prevarieties.- 2.3 Morphisms.- 2.4 Products.- 2.5 Hausdorff Axiom.- 3. Dimension.- 3.1 Dimension of a Variety.- 3.2 Dimension of a Subvariety.- 3.3 Dimension Theorem.- 3.4 Consequences.- 4. Morphisms.- 4.1 Fibres of a Morphism.- 4.2 Finite Morphisms.- 4.3 Image of a Morphism.- 4.4 Constructible Sets.- 4.5 Open Morphisms.- 4.6 Bijective Morphisms.- 4.7 Birational Morphisms.- 5. Tangent Spaces.- 5.1 Zariski Tangent Space.- 5.2 Existence of Simple Points.- 5.3 Local Ring of a Simple Point.- 5.4 Differential of a Morphism.- 5.5 Differential Criterion for Separability.- 6. Complete Varieties.- 6.1 Basic Properties.- 6.2 Completeness of Projective Varieties.- 6.3 Varieties Isomorphic to P1.- 6.4 Automorphisms of P1.- II. Affine Algebraic Groups.- 7. Basic Concepts and Examples.- 7.1 The Notion of Algebraic Group.- 7.2 Some Classical Groups.- 7.3 Identity Component.- 7.4 Subgroups and Homomorphisms.- 7.5 Generation by Irreducible Subsets.- 7.6 Hopf Algebras.- 8. Actions of Algebraic Groups on Varieties.- 8.1 Group Actions.- 8.2 Actions of Algebraic Groups.- 8.3 Closed Orbits.- 8.4 Semidirect Products.- 8.5 Translation of Functions.- 8.6 Linearization of Affine Groups.- III. Lie Algebras.- 9. Lie Algebra of an Algebraic Group.- 9.1 Lie Algebras and Tangent Spaces.- 9.2 Convolution.- 9.3 Examples.- 9.4 Subgroups and Lie Subalgebras.- 9.5 Dual Numbers.- 10. Differentiation.- 10.1 Some Elementary Formulas.- 10.2 Differential of Right Translation.- 10.3 The Adjoint Representation.- 10.4 Differential of Ad.- 10.5 Commutators.- 10.6 Centralizers.- 10.7 Automorphisms and Derivations.- IV. Homogeneous Spaces.- 11. Construction of Certain Representations.- 11.1 Action on Exterior Powers.- 11.2 A Theorem of Chevalley.- 11.3 Passage to Projective Space.- 11.4 Characters and Semi-Invariants.- 11.5 Normal Subgroups.- 12. Quotients.- 12.1 Universal Mapping Property.- 12.2 Topology of Y.- 12.3 Functions on Y.- 12.4 Complements.- 12.5 Characteristic 0.- V. Characteristic 0 Theory.- 13. Correspondence between Groups and Lie Algebras.- 13.1 The Lattice Correspondence.- 13.2 Invariants and Invariant Subspaces.- 13.3 Normal Subgroups and Ideals.- 13.4 Centers and Centralizers.- 13.5 Semisimple Groups and Lie Algebras.- 14. Semisimple Groups.- 14.1 The Adjoint Representation.- 14.2 Subgroups of a Semisimple Group.- 14.3 Complete Reducibility of Representations.- VI. Semisimple and Unipotent Elements.- 15. Jordan-Chevalley Decomposition.- 15.1 Decomposition of a Single Endomorphism.- 15.2 GL(n, K) and gl(n, K).- 15.3 Jordan Decomposition in Algebraic Groups.- 15.4 Commuting Sets of Endomorphisms.- 15.5 Structure of Commutative Algebraic Groups.- 16. Diagonalizable Groups.- 16.1 Characters and d-Groups.- 16.2 Tori.- 16.3 Rigidity of Diagonalizable Groups.- 16.4 Weights and Roots.- VII. Solvable Groups.- 17. Nilpotent and Solvable Groups.- 17.1 A Group-Theoretic Lemma.- 17.2 Commutator Groups.- 17.3 Solvable Groups.- 17.4 Nilpotent Groups.- 17.5 Unipotent Groups.- 17.6 Lie-Kolchin Theorem.- 18. Semisimple Elements.- 18.1 Global and Infinitesimal Centralizers.- 18.2 Closed Conjugacy Classes.- 18.3 Action of a Semisimple Element on a Unipotent Group.- 18.4 Action of a Diagonalizable Group.- 19. Connected Solvable Groups.- 19.1 An Exact Sequence.- 19.2 The Nilpotent Case.- 19.3 The General Case.- 19.4 Normalizer and Centralizer.- 19.5 Solvable and Unipotent Radicals.- 20. One Dimensional Groups.- 20.1 Commutativity of G.- 20.2 Vector Groups and e-Groups.- 20.3 Properties of p-Polynomials.- 20.4 Automorphisms of Vector Groups.- 20.5 The Main Theorem.- VIII. Borel Subgroups.- 21. Fi

User’s Reviews

Reviews from Amazon users which were colected at the time this book was published on the website:

⭐Recommend this book without reservation. an excellent book on the topics.

⭐I am currently using this text as an independent study in a graduate level seminar, covering the entirety in one semester. Thus far, we have covered the first ten chapters of this book, and have reached the following (unfortunately) unfavorable conclusion of this text.This text is relatively self-contained with fairly standard treatment of the subject of linear algebraic groups as varieties over an algebraic closed field (not necessarily characteristic 0). Despite being rooted in algebraic geometry, the subject has a fair mix of non-algebraic geometric arguments. Nonetheless, irreducibility, constructivility, finiteness and completeness are employed often. The scope is about comparable with Borel’s, and is a proper subset of TA Springer’s.I aim to talk about the author’s exposition. The structure of the book is very rigid. Chapters are organized into sections, and each into subsections. The content is very densely packed; significant points in the exposition are thereby difficult to parse out. At times, concepts were not realized to be significant until they were used often later in the text. Trivial points were belabored, notations were laid down carelessly, arguments were described and not detailed; understandable, though unpleasant.More than anything else that made this book difficult to read is that the proofs had numerous small to medium gaps. To fill those gaps, readers unfamiliar with the subject are left guessing blindly. Facts were stated as “obvious” or with very little guidance without explanation or reference to earlier results. These occur more or less frequently depending on how familiar one is with the subject. An early student of the subject may find that these gaps make progress unbearably slow. Without perspective, it is easy to miss the forest for the trees.To summarize the gripe — borrowing another tree analogy — reading Humphreys’s LAG is like hiking with all significant trail markers removed. Groping is a necessity; being lost a few times, a guarantee.One can interpret all this as a boon (perhaps with a felicitous application of Stockholm syndrome). The exercises are often simple, but to really read the text carefully, one must necessarily wield deftly a broad range of facts, and be able to conjure them from the slightest of clues. I believe this text is really suited for a month long exam for the student of linear algebraic groups; anyone who can read this book in one comfortable sitting must either know the subject warmly, or is truly equipped with the gift of learning.In any case, I recommend this not for even a determined self-student. When reading this text, it’s best to find a friendly expert, or try Borel’s, on which a large portion of this text is based. Borel is lengthier, scheme theoretic, and more classical. It will take about the same amount of time to read, but you will walk away from Borel believing more in yourself. For a mathematician, that may make the difference between a half cup of tea and a pitcher full of coffee.

⭐It is a book as it is

Keywords

Free Download Linear Algebraic Groups in PDF format
Linear Algebraic Groups PDF Free Download
Download Linear Algebraic Groups 2012 PDF Free
Linear Algebraic Groups 2012 PDF Free Download
Download Linear Algebraic Groups PDF
Free Download Ebook Linear Algebraic Groups