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Three Schur functors related to pre-Lie algebras

Part of: Homological algebra

Published online by Cambridge University Press:  16 October 2023

VLADIMIR DOTSENKO  and
OISÍN FLYNN-CONNOLLY
VLADIMIR DOTSENKO
Affiliation:
Institut de Recherche Mathématique Avancée, UMR 7501, Université de Strasbourg et CNRS, 7 rue René-Descartes, 67000 Strasbourg CEDEX, France. e-mail: vdotsenko@unistra.fr
OISÍN FLYNN-CONNOLLY
Affiliation:
Laboratoire de Géométrie, Analyse et Applications, UMR 7539, Université Sorbonne Paris Nord et CNRS, 93430, Villetaneuse, France. e-mail: flynn-connolly@math.univ-paris13.fr
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Abstract

We give explicit combinatorial descriptions of three Schur functors arising in the theory of pre-Lie algebras. The first of them leads to a functorial description of the underlying vector space of the universal enveloping pre-Lie algebra of a given Lie algebra, strengthening the Poincaré-Birkhoff-Witt (PBW) theorem of Segal. The two other Schur functors provide functorial descriptions of the underlying vector spaces of the universal multiplicative enveloping algebra and of the module of Kähler differentials of a given pre-Lie algebra. An important consequence of such descriptions is an interpretation of the cohomology of a pre-Lie algebra with coefficients in a module as a derived functor for the category of modules over the universal multiplicative enveloping algebra.

MSC classification

Secondary: 18G10: Resolutions; derived functors 18G15: Ext and Tor, generalizations, Künneth formula
Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 176 , Issue 2 , March 2024 , pp. 441 - 458
Copyright
© The Author(s), 2023. Published by Cambridge University Press on behalf of Cambridge Philosophical Society

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