Geodesic
Chain Realization of Differential Modules with $\infty$-Simplicial Faces and the $B$-Construction over $A_\infty$-Algebras
Matematičeskie zametki, Tome 98 (2015) no. 1, pp. 101-124 Cet article a éte moissonné depuis la source Math-Net.Ru

Voir la notice de l'article

On the basis of the colored version of Koszul duality, the notion of a differential module with $\infty$-simplicial faces is introduced. By using the homotopy technique of differential Lie modules over colored coalgebras, the homotopy invariance of the structure of a differential module with $\infty$-simplicial faces is proved. A relationship between differential modules with $\infty$-simplicial faces and $A_\infty$-algebras is described. The notions of the chain realization of a differential module with $\infty$-simplicial faces and the tensor product of differential modules with $\infty$-simplicial faces are introduced. It is shown that the chain realization of a tensor differential module with $\infty$-simplicial faces constructed from an $A_\infty$-algebra and the $B$-construction over this $A_\infty$-algebra are isomorphic differential coalgebras.
Keywords: differential module with $\infty$-simplicial faces, $A_\infty$-algebra, colored differential module, colored differential algebra, Koszul duality, chain realization of differential modules, category of differential Lie $C$-modules, SDR-data, differential $R_\infty$-module.
Mots-clés : $B$-construction 
@article{MZM_2015_98_1_a7,
     author = {S. V. Lapin},
     title = {Chain {Realization} of {Differential} {Modules} with $\infty${-Simplicial} {Faces} and the $B${-Construction} over $A_\infty${-Algebras}},
     journal = {Matemati\v{c}eskie zametki},
     pages = {101--124},
     year = {2015},
     volume = {98},
     number = {1},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/MZM_2015_98_1_a7/}
}
TY  - JOUR
AU  - S. V. Lapin
TI  - Chain Realization of Differential Modules with $\infty$-Simplicial Faces and the $B$-Construction over $A_\infty$-Algebras
JO  - Matematičeskie zametki
PY  - 2015
SP  - 101
EP  - 124
VL  - 98
IS  - 1
UR  - http://geodesic.mathdoc.fr/item/MZM_2015_98_1_a7/
LA  - ru
ID  - MZM_2015_98_1_a7
ER  - 
%0 Journal Article
%A S. V. Lapin
%T Chain Realization of Differential Modules with $\infty$-Simplicial Faces and the $B$-Construction over $A_\infty$-Algebras
%J Matematičeskie zametki
%D 2015
%P 101-124
%V 98
%N 1
%U http://geodesic.mathdoc.fr/item/MZM_2015_98_1_a7/
%G ru
%F MZM_2015_98_1_a7
S. V. Lapin. Chain Realization of Differential Modules with $\infty$-Simplicial Faces and the $B$-Construction over $A_\infty$-Algebras. Matematičeskie zametki, Tome 98 (2015) no. 1, pp. 101-124. http://geodesic.mathdoc.fr/item/MZM_2015_98_1_a7/

[1] V. A. Smirnov, “Gomotopicheskaya teoriya koalgebr”, Izv. AN SSSR. Ser. matem., 49:6 (1985), 1302–1321 | MR | Zbl

[2] V. A. Smirnov, “Gomologii $B$-konstruktsii i ko-$B$-konstruktsii”, Izv. RAN. Ser. matem., 58:4 (1994), 80–96 | MR | Zbl

[3] S. V. Lapin, “Prodolzhenie operatsii umnozheniya v $E_\infty$-algebrakh do $A_\infty$-morfizma $E_\infty$-algebr i kartanovskie ob'ekty v kategorii algebr Meya”, Matem. zametki, 89:5 (2011), 719–737 | DOI | MR | Zbl

[4] S. P. Novikov, “O kogomologiyakh algebry Stinroda”, Dokl. AN SSSR, 128:5 (1959), 893–895 | MR | Zbl

[5] J. D. Stasheff, “Homotopy associativity of $H$-spaces. I”, Trans. Amer. Math. Soc., 108:2 (1963), 275–292 | DOI | MR | Zbl

[6] T. V. Kadeishvili, “K teorii gomologii rassloennykh prostranstv”, UMN, 35:3 (1980), 183–188 | MR | Zbl

[7] V. A. Smirnov, Simplitsialnye i operadnye metody v algebraicheskoi topologii, Faktorial, M., 2002

[8] S. V. Lapin, “$D_\infty$-differentsialnye $A_\infty$-algebry i spektralnye posledovatelnosti”, Matem. sb., 193:1 (2002), 119–142 | DOI | MR | Zbl

[9] J.-L. Loday, B. Vallette, Algebraic Operads, Grundlehren Math. Wiss., 346, Springer-Verlag, Berlin, 2012 | DOI | MR | Zbl

[10] V. A. Smirnov, “Algebry Li nad operadami i ikh primenenie v teorii gomotopii”, Izv. RAN. Ser. matem., 62:3 (1998), 121–154 | DOI | MR | Zbl

[11] S. V. Lapin, “Differentsialnye moduli Li nad iskrivlennymi krashennymi koalgebrami i $\infty$-simplitsialnye moduli”, Matem. zametki, 96:5 (2014), 709–731 | DOI

[12] S. V. Lapin, “Gomotopicheskie simplitsialnye grani i gomologii realizatsii simplitsialnykh topologicheskikh prostranstv”, Matem. zametki, 94:5 (2013), 661–681 | DOI | MR | Zbl

[13] S. V. Lapin, “Gomotopicheskie svoistva differentsialnykh modulei Li nad iskrivlennymi koalgebrami i dvoistvennost po Koshulyu”, Matem. zametki, 94:3 (2013), 354–372 | DOI | MR | Zbl

[14] S. V. Lapin, “Differentsialnye vozmuscheniya i $D_\infty$-differentsialnye moduli”, Matem. sb., 192:11 (2001), 55–76 | DOI | MR | Zbl