QUANTUM gl ∞ , INFINITE q - SCHUR ALGEBRAS AND THEIR REPRESENTATIONS JIE

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) [1], 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 K̂ † (∞) so that ζr can be extended to an algebra epimorphism ζ̃r : K̂ † (∞) → S(∞, r). Finally we will investigate the representation theory of U(∞), especially the polynomial representations of U(∞). Dedicated to Professor Leonard L. Scott on the occasion of his 65th birthday