Controlling structures, deformations and homotopy theory for averaging algebras

Apurba Das

doi:10.5802/crmath.789

Article de recherche - Algèbre

Controlling structures, deformations and homotopy theory for averaging algebras
[Structures de contôle, déformations et théorie de l’homotopie pour les algèbres de moyenne]

Apurba Das ¹

¹Department of Mathematics, Indian Institute of Technology, Kharagpur 721302, West Bengal, India

Comptes Rendus. Mathématique, Volume 364 (2026), pp. 159-198

Abstract (VO)
Résumé

An averaging operator on an associative algebra $A$ is an algebraic abstraction of the time average operator on the space of real-valued functions defined in time-space. In this paper, we consider relative averaging operators on a bimodule $M$ over an associative algebra $A$. A relative averaging operator induces a diassociative algebra structure on the space $M$. The full data consisting of an associative algebra, a bimodule and a relative averaging operator is called a relative averaging algebra. We define bimodules over a relative averaging algebra that fits with the representations of diassociative algebras. We construct a graded Lie algebra and an $L_\infty $-algebra that are respectively controlling algebraic structures for a given relative averaging operator and relative averaging algebra. We also define cohomologies of relative averaging operators and relative averaging algebras and find a long exact sequence connecting various cohomology groups. As applications, we study deformations and abelian extensions of relative averaging algebras. Finally, we define homotopy relative averaging algebras and show that they induce homotopy diassociative algebras.

Un opérateur de moyenne sur une algèbre associative $A$ est une abstraction algébrique de l’opérateur de moyenne temporel sur l’espace des fonctions à valeurs réelles définies dans l’espace-temps. Dans cet article, nous considérons les opérateurs de moyenne relatifs sur un bimodule $M$ sur une algèbre associative $A$. Un opérateur de moyenne relatif induit une structure d’algèbre diassociative sur l’espace $M$. La donnée d’une algèbre associative, d’un bimodule et d’un opérateur de moyenne relatif est appelée algèbre de moyenne relative. Nous définissons des bimodules sur une algèbre de moyenne relative qui s’adaptent aux représentations des algèbres diassociatives. Nous construisons une algèbre de Lie graduée et une algèbre $L_\infty $ qui contrôlent les structures algébriques pour, respectivement, un opérateur de moyenne relatif et une algèbre de moyenne relative donnés. Nous définissons également la cohomologie des opérateurs de moyenne relatifs et des algèbres de moyenne relatives et trouvons une suite exacte longue reliant divers groupes de cohomologie. Comme application, nous étudions les déformations et les extensions abéliennes des algèbres de moyenne relatives. Enfin, nous définissons les algèbres de moyenne relatives homotopiques et montrons qu’elles induisent des algèbres diassociatives homotopiques.

Reçu le : 2024-03-04
Révisé le : 2024-12-03
Accepté le : 2025-08-22
Publié le : 2026-03-17

DOI : 10.5802/crmath.789

Classification : 16D20, 16W99, 16E40, 16S80
Keywords: Averaging algebras, diassociative algebras, $L_\infty $-algebras, deformations, homotopy structures
Mots-clés : Algèbres moyennes, algèbres diassociatives, algèbres $L_\infty $, déformations, structures d’homotopie

Affiliations des auteurs :

Apurba Das ¹

¹ Department of Mathematics, Indian Institute of Technology, Kharagpur 721302, West Bengal, India

Licence :

CC-BY 4.0

Droits d'auteur : Les auteurs conservent leurs droits

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     title = {Controlling structures, deformations and homotopy theory for averaging algebras},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {159--198},
     year = {2026},
     publisher = {Acad\'emie des sciences, Paris},
     volume = {364},
     doi = {10.5802/crmath.789},
     language = {en},
}

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Apurba Das. Controlling structures, deformations and homotopy theory for averaging algebras. Comptes Rendus. Mathématique, Volume 364 (2026), pp. 159-198. doi: 10.5802/crmath.789

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