Finite dimensional algebras and quantum groups

Contentsxi§12.3. Proof of Green’s formula 516§12.4. Green algebras and Lusztig’s theorem 523§12.5. Green’s theorem 527Exercises and notes 532Part 5. The BLM Algebra: A Realization for Quantum gl nChapter 13. Serre relations in quantum Schur algebras 537§13.1. n-step flags and the orbit–matrix correspondence 538§13.2. Dimensions of orbits 541§13.3. Orbits corresponding to almost diagonal matrices 544§13.4. A quantumization for quantum Schur algebras 546§13.5. The fundamental multiplication formulas 550§13.6. Some partial orderings on Ξ(n) and ˜Ξ(n) 558§13.7. The BLM triangular relations 560§13.8. Extending the fundamental multiplication formulas 567§13.9. Generators and relations 572§13.10. Presentations for quantum Schur algebras 577Exercises and notes 587Chapter 14. Constructing quantum gl n via quantum Schur algebras 591§14.1. A stabilization property 592§14.2. The BLM algebra K and its canonical basis 595§14.3. The completion ̂K of K and multiplication formulas 598§14.4. Embedding U v (gl n ) into ̂K 602§14.5. Z-forms of U v (gl n ) 606§14.6. Integral quantum Schur–Weyl reciprocity 609§14.7. A connection with Ringel–Hall algebras 614Exercises and notes 617AppendicesAppendix A. Varieties and affine algebraic groups 623§A.1. Affine varieties 624§A.2. Varieties 630§A.3. Affine algebraic groups 633§A.4. Parabolic subgroups and the Chevalley–Bruhat ordering 643§A.5. Representation theory: a first view 645