Quantum toroidal algebras and motivic Hall algebras I. Hall algebras for singular elliptic curves

TL;DR

This paper explores the structure of motivic Hall algebras over singular elliptic curves, establishing their isomorphism with smooth cases and connecting them to quantum toroidal algebras, revealing algebraic automorphisms.

Contribution

It introduces the composition subalgebra for singular elliptic curves and proves its isomorphism to the smooth case, linking Hall algebras to quantum toroidal algebras and automorphisms.

Findings

The composition subalgebra for singular curves is isomorphic to that of smooth elliptic curves.

The reduced Drinfeld double corresponds to the quantum toroidal algebra for rak{gl}_1.

An automorphism matches Miki's algebraic construction.

Abstract

We consider the motivic Hall algebra of coherent sheaves over an irreducible reduced projective curve of arithmetic genus $1$. We introduce the composition subalgebra in the singular curve case, and show that it is isomorphic to the composition subalgebra for a smooth elliptic curve. As in the case of smooth elliptic case studied by Burban and Schiffmann, the reduced Drinfeld double of the composition subalgebra is isomorphic to the quantum toroidal algebra for $\mathfrak{gl}_1$ (also called Ding-Iohara-Miki algebra), and it inherits automorphisms induced from equivalences of the associated derived category. We show that one of the non-trivial automorphisms coincide with the one constructed by Miki in a purely algebraic manner.