Quantum affine [formula omitted] via Hecke algebras

Abstract

The quantum loop algebra of gln is the affine analogue of quantum gln. In the seminal work [1], Beilinson–Lusztig–MacPherson gave a beautiful realisation for quantum gln 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 gln. 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 gln, our approach is purely algebraic and combinatorial, independent of the geometric method for quantum gln. 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.