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Stable homology of Lie algebras of derivations and homotopy invariants of wheeled operads

Part of: Lie algebras and Lie superalgebras Higher algebraic $K$-theory General nonassociative rings

Published online by Cambridge University Press:  13 August 2025

Vladimir Dotsenko Open the ORCID record for Vladimir Dotsenko [Opens in a new window]
Vladimir Dotsenko*
Affiliation:
Institut de Recherche Mathématique Avancée, UMR 7501, Université de Strasbourg et CNRS, 7 rue René-Descartes, 67000 Strasbourg CEDEX, France vdotsenko@unistra.fr
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Abstract

We prove a theorem that computes, for any augmented operad $\mathcal{O}$, the stable homology of the Lie algebra of derivations of the free algebra $\mathcal{O}(V)$ with twisted bivariant coefficients (here stabilization occurs as $\dim(V)\to\infty$) out of the homology of the wheeled bar construction of $\mathcal{O}$; this can further be used to prove uniform mixed representation stability for the homology of the positive part of that Lie algebra with constant coefficients. This result generalizes both the Loday–Quillen–Tsygan theorem on the homology of the Lie algebra of infinite matrices and the Fuchs stability theorem for the homology of the Lie algebra of vector fields. We also prove analogous theorems for the Lie algebras of derivations with constant and zero divergence, in which case one has to consider the wheeled bar construction of the wheeled completion of $\mathcal{O}$. Similarly to how cyclic homology of an algebra A may be viewed as an additive version of the algebraic K-theory of A, our results hint at the additive K-theoretic nature of the wheeled bar construction.

Keywords

cyclic homologygraph complexLie algebra homologywheeled operad

MSC classification

Primary: 17B56: Cohomology of Lie (super)algebras
Secondary: 17A36: Automorphisms, derivations, other operators 19D55: $K$-theory and homology; cyclic homology and cohomology

Information

Type
Research Article
Information
Compositio Mathematica , Volume 161 , Issue 4 , April 2025 , pp. 756 - 799
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Copyright
© The Author(s), 2025. The publishing rights in this article are licensed to Foundation Compositio Mathematica under an exclusive licence

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Footnotes

To Boris Feigin, with gratitude and admiration.

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