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

TL;DR

This paper describes the algebraic structure of the K-theory of semi-nilpotent commuting varieties of gl_n, providing explicit shuffle algebra and generator-based descriptions of its convolution algebra structure.

Contribution

It introduces a novel explicit shuffle algebra description and generator-based presentation of the convolution algebra structure in the K-theory of semi-nilpotent commuting varieties.

Findings

Explicit shuffle algebra description of the convolution algebra.

Generator set for the algebra involving elements H_{n,d}.

Connection between geometric K-theory and algebraic structures.

Abstract

We consider the (direct sum over all $n$ of the) $K$-theory of the semi-nilpotent commuting variety of $\mathfrak{gl}_n$, 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 $\{\bar{H}_{n,d}\}_{(n,d) \in \mathbb{N} \times \mathbb{Z}}$.