A Tour of Representation Theory (Graduate Studies in Mathematics) (Graduate Studies in Mathematics, 193)

"Representation theory investigates the different ways in which a given algebraic object such as a group or a Lie algebra can act on a vector space. Besides being a subject of great intrinsic beauty, the theory enjoys the additional benefit of having applications in myriad contexts outside pure mathematics, including quantum field theory and the study of molecules in chemistry. Adopting a panoramic viewpoint, this book offers an introduction to four different flavors of representation theory: representations of algebras, groups, Lie algebras, and Hopf algebras. A separate part of the book is devoted to each of these areas and they are all treated in sufficient depth to enable and hopefully entice the reader to pursue research in representation theory. The book is intended as a textbook for a course on representation theory, which could immediately follow the standard graduate abstract algebra course, and for subsequent more advanced reading courses. Therefore, more than 350 exercises at various levels of difficulty are included. The broad range of topics covered will also make the text a valuable reference for researchers in algebra and related areas and a source for graduate and postgraduate students wishing to learn more about representation theory by self-study." -- Provided by publisher Cover Contents Preface Conventions Part I. Algebras Chapter 1 Representations of Algebras 1.1 Algebras 1.2 Representations 1.3 Primitive Ideals 1.4 Semisimplicity 1.5 Characters Chapter 2 Further Topics on Algebras 2.1 Projectives 2.2 Frobenius and Symmetric Algebras Part II. Groups Chapter 3 Groups and Group Algebras 3.1 Generalities 3.2 First Examples 3.3 More Structure 3.4 Semisimple Group Algebras 3.5 Further Examples 3.6 Some Classical Theorems 3.7 Characters, Symmetric Polynomials, and Invariant Theory 3.8 Decomposing Tensor Powers Chapter 4 Symmetric Groups 4.1 Gelfand-Zetlin Algebras 4.2 The Branching Graph 4.3 The Young Graph 4.4 Proof of the Graph Isomorphism Theorem 4.5 The Irreducible Representations 4.6 The Murnaghan-Nakayama Rule 4.7 Schur-Weyl Duality Part III. Lie Algebras Chapter 5 Lie Algebras and Enveloping Algebras 5.1 Lie Algebra Basics 5.2 Types of Lie Algebras 5.3 Three Theorems about Linear Lie Algebras 5.4 Enveloping Algebras 5.5 Generalities on Representations of Lie Algebras 5.6 The Nullstellensatz for Enveloping Algebras 5.7 Representations of sl_2 Chapter 6 Semisimple Lie Algebras 6.1 Characterizations of Semisimplicity 6.2 Complete Reducibility 6.3 Cartan Subalgebras and the Root Space Decomposition 6.4 The Classical Lie Algebras Chapter 7 Root Systems 7.1 Abstract Root Systems 7.2 Bases of a Root System 7.3 Classification 7.4 Lattices Associated to a Root System Chapter 8 Representations of Semisimple Lie Algebras 8.1 Reminders 8.2 Finite-Dimensional Representations 8.3 Highest Weight Representations 8.4 Finite-Dimensional Irreducible Representations 8.5 The Representation Ring 8.6 The Center of the Enveloping Algebra 8.7 Weyl’s Character Formula 8.8 Schur Functors and Representations of sl(V) Part IV. Hopf Algebras Chapter 9 Coalgebras, Bialgebras, and Hopf Algebras 9.1 Coalgebras 9.2 Comodules 9.3 Bialgebras and Hopf Algebras Chapter 10 Representations and Actions 10.1 Representations of Hopf Algebras 10.2 First Applications 10.3 The Representation Ring of a Hopf Algebra 10.4 Actions and Coactions of Hopf Algebras on Algebras Chapter 11 Affine Algebraic Groups 11.1 Affine Group Schemes 11.2 Affine Algebraic Groups 11.3 Representations and Actions 11.4 Linearity 11.5 Irreducibility and Connectedness 11.6 The Lie Algebra of an Affine Algebraic Group 11.7 Algebraic Group Actions on Prime Spectra Chapter 12 Finite-Dimensional Hopf Algebras 12.1 Frobenius Structure 12.2 The Antipode 12.3 Semisimplicity 12.4 Divisibility Theorems 12.5 Frobenius-Schur Indicators Appendices Appendix A. The Language of Categories and Functors A.1 Categories A.2 Functors A.3 Naturality A.4 Adjointness Appendix B. Background from Linear Algebra B.1 Tensor Products B.2 Hom-⊗ Relations B.3 Vector Spaces Appendix C. Some Commutative Algebra C.1 The Nullstellensatz C.2 The Generic Flatness Lemma C.3 The Zariski Topology on a Vector Space Appendix D. The Diamond Lemma D.1 The Goal D.2 The Method D.3 First Applications D.4 A Simplification D.5 The Poincaré-Birkhoff-Witt Theorem Appendix E. The Symmetric Ring of Quotients E.1 Definition and Basic Properties E.2 The Extended Center E.3 Comparison with Other Rings of Quotients Bibliography Subject Index Index of Names Notation

Representation theory investigates the different ways in which a given algebraic object—such as a group or a Lie algebra—can act on a vector space. Besides being a subject of great intrinsic beauty, the theory enjoys the additional benefit of having applications in myriad contexts outside pure mathematics, including quantum field theory and the study of molecules in chemistry.Adopting a panoramic viewpoint, this book offers an introduction to four different flavors of representation theory: representations of algebras, groups, Lie algebras, and Hopf algebras. A separate part of the book is devoted to each of these areas and they are all treated in sufficient depth to enable and hopefully entice the reader to pursue research in representation theory.The book is intended as a textbook for a course on representation theory, which could immediately follow the standard graduate abstract algebra course, and for subsequent more advanced reading courses. Therefore, more than 350 exercises at various levels of difficulty are included. The broad range of topics covered will also make the text a valuable reference for researchers in algebra and related areas and a source for graduate and postgraduate students wishing to learn more about representation theory by self-study.

Offers an introduction to four different flavours of representation theory: representations of algebras, groups, Lie algebras, and Hopf algebras. A separate part of the book is devoted to each of these areas and they are all treated in sufficient depth to enable the reader to pursue research in representation theory.