A Double Hall Algebra Approach to Affine Quantum Schur–Weyl Theory

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.

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.