Quantum gl∞, infinite q-Schur algebras and their representations

In this paper, we investigate the structure and representations of the quantum group U(∞)=Uυ(gl∞). We will present a realization for U(∞), following Beilinson–Lusztig–MacPherson (BLM) [A.A. Beilinson, G. Lusztig, R. MacPherson, A geometric setting for the quantum deformation of GLn, Duke Math. J. 61 (1990) 655–677], and show that the natural algebra homomorphism ζr from U(∞) to the infinite q-Schur algebra S(∞,r) is not surjective for any r⩾1. We will give a BLM type realization for the image U(∞,r):=ζr(U(∞)) and discuss its presentation in terms of generators and relations. We further construct a certain completion algebra so that ζr can be extended to an algebra epimorphism . Finally we will investigate the representation theory of U(∞), especially the polynomial representations of U(∞).