Hostname: page-component-6898cfcb55-kwdml Total loading time: 0 Render date: 2026-11-11T07:13:46.772Z Has data issue: false hasContentIssue false
  • English
  • Français

DIVIDED POWER ALGEBRAS OVER AN OPERAD

Part of:General nonassociative ringsCategories with structure

Published online by Cambridge University Press:  27 May 2019

SACHA IKONICOFF
SACHA IKONICOFF*
Affiliation:
Univ Paris Diderot, Institut de Mathématiques de Jussieu-Paris Rive Gauche, CNRS, Sorbonne Université, 8 place Aurélie Nemours, F-75013 Paris, France e-mail: sacha.ikonicoff@imj-prg.fr
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the 'Save PDF' action button.

The purpose of this paper is to give a characterisation of divided power algebras over a reduced operad. Such a characterisation is given in terms of polynomial operations, following the classical example of divided power algebras. We describe these polynomial operations in two different ways: one way uses invariant elements under the action of the symmetric group and the other coinvariant elements. Our results are then applied to the case of level algebras, which are (non-associative) commutative algebras satisfying the exchange law.

MSC classification

Primary:18D50: Operads
Secondary:17A30: Algebras satisfying other identities

Information

Type
Research Article
Information
Glasgow Mathematical Journal , Volume 62 , Issue 3 , September 2020, pp. 477 - 517
Copyright
© Glasgow Mathematical Journal Trust 2019

References

Séminaire Henri Cartan de l’Ecole Normale Supérieure, 1954/1955, Algèbres d’Eilenberg-MacLane et homotopie, Secrétariat mathématique, 11 rue Pierre Curie, Paris, 1956. 2ème éd.Google Scholar
Batanin, M. A. and Berger, C., The lattice path operad and Hochschild cochains, in Alpine perspectives on algebraic topology, vol. 504 of Contemp. Math. (American Mathematical Society, Providence, RI, 2009), 23–52.CrossRefGoogle Scholar
Gunnar Carlsson, G. B. Segal’s Burnside ring conjecture for (Z/2)k, Topology, 22(1) (1983), 83–103.CrossRefGoogle Scholar
Cesaro, Andrea, Pre-Lie algebras and operads in positive characteristic, PhD Thesis, (Thèse de doctorat dirigée par Fresse, Benoît et Vespa, Christine Mathématiques pures Lille 1, 2016).Google Scholar
Chataur, David and Livernet, Muriel, Adem-cartan operads, Commun. Algeb. 33(33) (2005), 4337–4360.CrossRefGoogle Scholar
Davis, Donald M., A family of unstable steenrod-modules which includes those of g. carlsson, J. Pure App. Algeb. 35 (1985), 253–267.CrossRefGoogle Scholar
Dokas, I., Pre-Lie algebras in positive characteristic, J. Lie Theor. 23(23) (2013), 937–952.Google Scholar
Elsholtz, Christian, Heuberger, Clemens, and Prodinger, Helmut, The number of Huffman codes, compact trees, and sums of unit fractions, IEEE Trans. Inform. Theor. 59(59) (2013), 1065–1075.CrossRefGoogle Scholar
Fresse, Benoit, On the homotopy of simplicial algebras over an operad, Trans. Amer. Math. Soc. 352(352) (2000), 4113–4141.CrossRefGoogle Scholar
Loday, Jean-Louis and Vallette, Bruno, Algebraic operads, vol. 346 of Grundlehren der Mathematischen Wissenschaften [Fundamental principles of mathematical sciences] (Springer, Heidelberg, 2012).Google Scholar
Nagpal, Rohit and Snowden, Andrew, Periodicity in the cohomology of symmetric groups via divided powers, Proc. Lond. Math. Soc. 116(5), 1244–1268.CrossRefGoogle Scholar
Richter, Birgit, Divided power structures and chain complexes, in Alpine perspectives on algebraic topology, vol. 504 of Contemp. Math. (American Mathematical Society, Providence, RI, 2009), 237–254.CrossRefGoogle Scholar
Roby, Norbert, Les algèbres à puissances divisées, Bull. Sci. Math. 89 (2) (1965), 75–91.Google Scholar
Schwartz, Lionel, Unstable modules over the Steenrod algebra and Sullivan ’s fixed point set conjecture, Chicago Lectures in Mathematics (University of Chicago Press, Chicago, IL, 1994).Google Scholar