Algebras, Rings and Modules: Lie Algebras and Hopf Algebras (Mathematical Surveys and Monographs)

The main goal of this book is to present an introduction to and applications of the theory of Hopf algebras. The authors also discuss some important aspects of the theory of Lie algebras. The first chapter can be viewed as a primer on Lie algebras, with the main goal to explain and prove the Gabriel-Bernstein-Gelfand-Ponomarev theorem on the correspondence between the representations of Lie algebras and quivers; this material has not previously appeared in book form. The next two chapters are also ''primers'' on coalgebras and Hopf algebras, respectively; they aim specifically to give sufficient background on these topics for use in the main part of the book. Chapters 4-7 are devoted to four of the most beautiful Hopf algebras currently known: the Hopf algebra of symmetric functions, the Hopf algebra of representations of the symmetric groups (although these two are isomorphic, they are very different in the aspects they bring to the forefront), the Hopf algebras of the nonsymmetric and quasisymmetric functions (these two are dual and both generalize the previous two), and the Hopf algebra of permutations. The last chapter is a survey of applications of Hopf algebras in many varied parts of mathematics and physics. Unique features of the book include a new way to introduce Hopf algebras and coalgebras, an extensive discussion of the many universal properties of the functor of the Witt vectors, a thorough discussion of duality aspects of all the Hopf algebras mentioned, emphasis on the combinatorial aspects of Hopf algebras, and a survey of applications already mentioned. The book also contains an extensive (more than 700 entries) bibliography Machine Generated Contents Note: Ch. 1 Lie Algebras And Dynkin Diagrams -- 1.1. Lie Algebras. Definitions And Examples -- 1.2. Ideals, Homomorphisms And Representations -- 1.3. Solvable And Nilpotent Lie Algebras -- 1.4. Radical Of A Lie Algebra. Simple And Semisimple Lie Algebras -- 1.5. Modules For Lie Algebras. Weyl's Theorem. Ado's Theorem -- 1.6. Lie's Theorem -- 1.7. Lie Algebra Sl(2;fc). Representation Of Sl(2;f)c -- 1.8. Universal Enveloping Algebra Of A Lie Algebra -- 1.9. Poincare-birkhoff-witt Theorem -- 1.10. Free Lie Algebras -- 1.11. Examples Of Simple Lie Algebras -- 1.12. Abstract Root Systems And The Weyl Group -- 1.13. Cartan Matrices And Dynkin Diagrams -- 1.14. Coxeter Groups And Coxeter Diagrams -- 1.15. Root Systems Of Semisimple Lie Algebras -- 1.16. Weyl Group Of A Quiver -- 1.17. Reflection Functors -- 1.18. Coxeter Functors And Coxeter Transformations -- 1.19. Gabriel Theorem -- 1.20. Generalized Cartan Matrices And Kac-moody Lie Algebras -- 1.21. Historical Notes -- References -- Ch. 2 Coalgebras: Motivation, Definitions, And Examples -- 2.1. Coalgebras And `addition Formulae' -- 2.2. Coalgebras And Decompositions -- 2.3. Dualizing The Idea Of An Algebra -- 2.4. Some Examples Of Coalgebras -- 2.5. Sub Coalgebras And Quotient Coalgebras -- 2.6. Main Theorem Of Coalgebras -- 2.7. Cofree Coalgebras -- 2.8. Algebra -- Coalgebra Duality -- 2.9. Comodules And Representations -- 2.10. Graded Coalgebras -- 2.11. Reflexive Modules -- 2.12. Measuring -- 2.13. Addition Formulae And Duality -- 2.14. Coradical And Coradical Filtration -- 2.15. Coda To Chapter 2 -- References -- Ch. 3 Bialgebras And Hopf Algebras. Motivation, Definitions, And Examples -- 3.1. Products And Representations -- 3.2. Bialgebras -- 3.3. Hopf Algebras -- 3.4. Some More Examples Of Hopf Algebras -- 3.5. Primitive Elements -- 3.6. Group-like Elements -- 3.7. Bialgebra And Hopf Algebra Duality -- 3.8. Graded Bialgebras And Hopf Algebras -- 3.9. Crossed Products -- 3.10. Integrals For Hopf Algebras -- 3.11. Formal Groups -- 3.12. Hopf Modules -- 3.13. Historical Remarks -- 3.14. Hopf Algebra Of An Algebra -- References -- Ch. 4 Hopf Algebra Of Symmetric Functions -- 4.1. Algebra Of Symmetric Functions -- 4.2. Hopf Algebra Structure -- 4.3. Psh Algebras -- 4.4. Automorphisms Of Symm -- 4.5. Functor Of The Witt Vectors -- 4.6. Ghost Components -- 4.7. Frobenius And Verschiebung Endomorplisms -- 4.8. Second Multiplication Of Symm -- 4.9. Lambda Algebras -- 4.10. Exp Algebras -- 4.11. Plethysm -- 4.12. Many Incarnations Of Symm -- References -- Ch. 5 Representations Of The Symmetric Groups From The Hopf Algebra Point Of View -- 5.1. Little Bit Of Finite Group Representation Theory -- 5.2. Double Cosets Of Young Subgroups -- 5.3. Hopf Algebra -- 5.4. Symm As A Psh Algebra -- 5.5. Second Multiplication On Rs -- 5.6. Remarks And Acknowledgements -- References -- Ch. 6 Hopf Algebra Of Noncommutative Symmetric Functions And The Hopf Algebra Of Quasisymmetric Functions -- 6.1. Hopf Algebra Nsymm -- 6.2. Nsymm Over The Rationals -- 6.3. Hopf Algebra Qsymm -- 6.4. Symm As A Quotient Of Nsymm -- 6.5. More On Shuffle And Liehopf -- 6.6. Autoduality Of Symm -- 6.7. Polynomial Freeness Of Qsymm Over The Integers -- 6.8. Hopf Endomorphisms Of Qsymm And Nsymm -- 6.9. Verschiebung And Frobenius On Nsymm And Qsymm -- References -- Ch. 7 Hopf Algebra Of Permutations -- 7.1. Hopf Algebra Of Permutations Of Malvenuto, Poirier And Reutenauer -- 7.2. Imbedding Of Nsymm Into Mpr -- 7.3. Lsd Permutations -- 7.4. Second Multiplication And Second Comultiplication -- 7.5. Rigidity And Uniqueness Of Mpr -- References -- Ch. 8 Hopf Algebras: Applications In And Interrelations With Other Parts Of Mathematics And Physics -- 8.1. Actions And Coactions Of Bialgebras And Hopf Algebras -- 8.2. Quantum Groups Glq(n, C) And Multiparameter Generalizations -- 8.3. Quantum Groups Uq(sl(n; K)) -- 8.4. R-matrices And Qist: Bethe Ansatz, Fcr Construction And The Frt Theorem -- 8.5. Knot Invariants From Quantum Groups. Yang-baxter Operators -- 8.6. Quiver Hopf Algebras -- 8.7. Ringel-hall Hopf Algebras -- 8.8. More Interactions Of Hopf Algebras With Other Parts Of Mathematics And Theoretical Physics -- 8.8.1. Capelli Identities And Other Formulas For Matrices And Determinants With Mildly Noncommuting Entries -- 8.8.2. Quantum Symmetry -- 8.8.3. Hopf Algebra Symmetries In Noncommutative Geometry -- 8.8.4. Hopf Algebras In Galois Theory -- 8.8.5. Hopf Algebras And Renormalization -- 8.8.6. Quantum Calculi -- 8.8.7. Umbral Calculus And Baxter Algebras -- 8.8.8. Q-special Functions -- References. Michiel Hazewinkel, Nadiya Gubareni, V.v. Kirichenko. Includes Bibliographical References And Index. Presenting an introduction to the theory of Hopf algebras, the authors also discuss some important aspects of the theory of Lie algebras. This book includes a chapters on the Hopf algebra of symmetric functions, the Hopf algebra of representations of the symmetric groups, the Hopf algebras of the nonsymmetric and quasisymmetric functions, and the Hopf algebra of permutations.