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The inertia operator on the motivic Hall algebra

Part of: Families, fibrations

Published online by Cambridge University Press:  14 March 2019

Kai Behrend  and
Pooya Ronagh
Kai Behrend
Affiliation:
Department of Mathematics, University of British Columbia, 1984 Mathematics Road, Vancouver, British Columbia, V6T 1Z2, Canada email behrend@math.ubc.ca
Pooya Ronagh
Affiliation:
Institute for Quantum Computing, University of Waterloo, 200 University Avenue West, Waterloo, Ontario, N2L 3G1, Canada email pooya.ronagh@uwaterloo.ca
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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.

Keywords

algebraic stacksinertia stackmotivic Hall algebraDonaldson–Thomas theory

MSC classification

Primary: 14D23: Stacks and moduli problems
Type
Research Article
Information
Compositio Mathematica , Volume 155 , Issue 3 , March 2019 , pp. 528 - 598
Copyright
© The Authors 2019 

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