The inertia operator on the motivic Hall algebra

TL;DR

This paper investigates the inertia operator in the motivic Hall algebra, demonstrating its diagonalizability and establishing a filtration that reveals a Lie algebra structure of virtually indecomposable elements.

Contribution

It introduces the concept of algebroids and proves the diagonalizability of the inertia operator, leading to new structural insights into the motivic Hall algebra.

Findings

Inertia operator is diagonalizable.

Filtration of the Hall algebra with a commutative associated graded.

Lie algebra of virtually indecomposable elements.

Abstract

We study the action of the inertia operator on the motivic Hall algebra, and prove that it is diagonalizable. This leads to a filtration of the Hall algebra, whose associated graded algebra is commutative. In particular, the degree 1 subspace forms a Lie algebra, which we call the Lie algebra of virtually indecomposable elements, following Joyce. We prove that the integral of virtually indecomposable elements admits an Euler characteristic specialization. In order to take advantage of the fact that our inertia groups are unit groups in algebras, we introduce the notion of algebroid.