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1 results

  • The inertia operator on the motivic Hall algebra

  • Part of
  • Kai Behrend, Pooya Ronagh
  • Journal:
    Compositio Mathematica / Volume 155 / Issue 3 / March 2019
    Published online by Cambridge University Press:
    14 March 2019, pp. 528-598
    Print publication:
    March 2019

  • 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.