AN INTEGRAL FORM OF QUANTUM TOROIDAL gl1

Abstract

We consider the (direct sum over all n ∈ N of the) K-theory of the seminilpotent commuting variety of gln, and describe its convolution algebra structure in two ways: the first as an explicit shuffle algebra (i.e., a particular Z[q±1 1 , q±1 2 ]-submodule of the equivariant K-theory of a point) and the second as the Z[q±1 1 , q±1 2 ]-algebra generated by certain elements { ¯Hn,d}(n,d)∈N×Z. As the shuffle algebra over Q(q1, q2) has long been known to be isomorphic to half of an algebra known as quantum toroidal gl1, we thus obtain a description of an important integral form of the quantum toroidal algebra.