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The inertia operator on the motivic Hall algebra

2019; Cambridge University Press; Volume: 155; Issue: 3 Linguagem: Inglês

10.1112/s0010437x18007881

1570-5846

Kai Behrend, Pooya Ronagh,

Algebraic Geometry and Number Theory

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.