An integral form of quantum toroidal ${\mathfrak{g}\mathfrak{l}}_1$

We consider the (direct sum over all $n$ of the) $K$-theory of the semi-nilpotent commuting variety of ${\mathfrak{g}\mathfrak{l}}_n$, and describe its convolution algebra structure in two ways: the first as an explicit shuffle algebra (i.e. a particular $\mathbb{Z}[q_1^{\pm 1},q_2^{\pm 1}]$-submodule of the equivariant $K$-theory of a point) and the second as the $\mathbb{Z}[q_1^{\pm 1},q_2^{\pm 1}]$-algebra generated by certain elements $\{{\bar{H}}_{n,d}{\}}_{(n,d)\in \mathbb{N}\times \mathbb{Z}}$.