Quantum affine gln via Hecke algebras

The quantum loop algebra of gl(n), is the affine analogue of quantum gl(n). In the seminal work [1], Beilinson-Lusztig-MacPherson gave a beautiful realisation for quantum gl(n), via a geometric setting of quantum Schur algebras. More precisely, they used quantum Schur algebras to construct a certain algebra U in [1, 5.4] and proved in [1, 5.7] that U is isomorphic to quantum gl(n). We will present in this paper a full generalisation of BLM's realisation to the affine case. Though the realisation of the quantum loop algebra of gln is motivated by the work [1] for quantum gl(n), our approach is purely algebraic and combinatorial, independent of the geometric method for quantum gl(n). As an application, we discover a presentation of the Ringel-Hall algebra of a cyclic quiver by semisimple generators and their multiplications by the defining basis elements. (C) 2015 Elsevier Inc. All rights reserved.