A Double Hall Algebra Approach to Affine Quantum Schur–Weyl Theory
London Mathematical Society Lecture Note Series

The theory of Schur?Weyl duality has had a profound influence over many areas of algebra and combinatorics. This text is original in two respects: it discusses affine q-Schur algebras and presents an algebraic, as opposed to geometric, approach to affine quantum Schur?Weyl theory. To begin, various algebraic structures are discussed, including double Ringel?Hall algebras of cyclic quivers and their quantum loop algebra interpretation. The rest of the book investigates the affine quantum Schur?Weyl duality on three levels. This includes the affine quantum Schur?Weyl reciprocity, the bridging role of affine q-Schur algebras between representations of the quantum loop algebras and those of the corresponding affine Hecke algebras, presentation of affine quantum Schur algebras and the realisation conjecture for the double Ringel?Hall algebra with a proof of the classical case. This text is ideal for researchers in algebra and graduate students who want to master Ringel?Hall algebras and Schur?Weyl duality.

Introduction; 1. Preliminaries; 2. Double Ringel–Hall algebras of cyclic quivers; 3. Affine quantum Schur algebras and the Schur–Weyl reciprocity; 4. Representations of affine quantum Schur algebras; 5. The presentation and realization problems; 6. The classical (v =1) case; Bibliography; Index.

Dr Bangming Deng is a Professor of Mathematics at Beijing Normal University, having previously been an Alexander von Humboldt Fellow at Bielefeld University. His main research interests are representation theory of algebras, Ringel–Hall algebras, Lie algebras and quantum groups. He is a co-author of the book Finite Dimensional Algebras and Quantum Groups (2008) and has served as editor for the Journal of Frontiers of Mathematics in China.
Dr Jie Du has taught at the University of Virginia, the University of Sydney and is currently an Associate Professor at the University of New South Wales, Sydney. His main research areas are Lie theory, representation theory and related algebras. He is a co-author of the book Finite Dimensional Algebras and Quantum Groups (2008) and has served as section editor for the Journal of the Australian Mathematical Society and the Bulletin of the Malaysian Mathematical Science Society.
Dr Qiang Fu is a Professor at Tongji University in Shanghai. His interests lie in the representation theories of algebraic and quantum groups, finite dimensional algebras and related topics. His recent work has concentrated mainly on little q-Schur algebras, infinitesimal q-Schur algebras, affine q-Schur algebras and quantum affine gln.