Elliptic Hall algebra on $\mathbb{F}_1$

TL;DR

This paper constructs an elliptic Hall algebra over the field with one element using monoidal schemes and monoidal representations, linking it to known elliptic Hall and skein algebras.

Contribution

It introduces a novel construction of elliptic Hall algebra over 1, connecting monoidal scheme theory with Hall algebra and skein algebra frameworks.

Findings

The algebra is a specialization of Burban and Schiffmann's elliptic Hall algebra.

It is isomorphic to the skein algebra for the torus.

The construction bridges monoidal schemes with algebraic structures.

Abstract

We construct Hall algebra of elliptic curve over $\mathbb{F}_1$ using the theory of monoidal scheme due to Deitmar and the theory of Hall algebra for monoidal representations due to Szczesny. The resulting algebra is shown to be a specialization of elliptic Hall algebra studied by Burban and Schiffmann. Thus our algebra is isomorphic to the skein algebra for torus by the recent work of Morton and Samuelson.