×

zbMATH — the first resource for mathematics

Cup \(i\)-product on the graded algebras with symmetries and Gerstenhaber algebra. (Cup \(i\)-produit sur les algèbres graduées avec symétries et algèbres de Gerstenhaber.) (French. English summary) Zbl 1291.18024
In this work, the author defines the notion of cup \(i\)-products on the graded algebras with symmetries. She proves a nice relation between these cup \(i\)-products and the differential forms of the graded algebras with symmetries. These algebras are a generalization of the noncommutative differential forms algebras which were defined by A. Connes [Publ. Math., Inst. Hautes Étud. Sci. 62, 41–144 (1985; Zbl 0592.46056)]. The author proves that the cohomology of a graded algebra with symmetries is a Gerstenhaber algebra.
MSC:
18G55 Nonabelian homotopical algebra (MSC2010)
58A10 Differential forms in global analysis
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] Battikh, N., Cup \(i\)-produits sur LES formes différentielles non commutatives et carrés de Steenrod, Journal of Algebra, 313, 531-553, (2007) · Zbl 1117.55015
[2] Battikh, N., Algèbres graduées avec symétries, Journal of Algebra, 325, 49-73, (2011) · Zbl 1238.16008
[3] Connes, A., Non commutative différential geometry, Pub. Math. I.H.E.S, 62, 257-360, (1985) · Zbl 0592.46056
[4] Cuntz, Joachim; Quillen, Daniel, Cyclic homology and nonsingularity, J. Amer. Math. Soc., 8, 2, 373-442, (1995) · Zbl 0838.19002
[5] Dold, Albrecht; Thom, René, Une généralisation de la notion d’espace fibré. application aux produits symétriques infinis, C. R. Acad. Sci. Paris, 242, 1680-1682, (1956) · Zbl 0071.17301
[6] Gerstenhaber, M., The cohomology structure of an associative ring, Ann of Math., 78, 267-288, (1963) · Zbl 0131.27302
[7] Karoubi, Max, Formes différentielles non commutatives et cohomology à coefficients arbitraires, Transaction of the AMS, 347, 4277-4299, (1995) · Zbl 0852.55009
[8] Karoubi, Max, Formes topologiques non commutatives, Annales scientifiques. E. N. S., 28, 477-492, (1995) · Zbl 0837.55004
[9] Kraines, David, Massey higher products, Trans. Amer. Math. Soc., 124, 431-449, (1966) · Zbl 0146.19201
[10] May, J. Peter, The Steenrod Algebra and its Applications (Proc. Conf. to Celebrate N. E. Steenrod’s Sixtieth Birthday, Battelle Memorial Inst., Columbus, Ohio, 1970), A general algebraic approach to Steenrod operations, 153-231, (1970), Springer, Berlin · Zbl 0242.55023
[11] Steenrod, N. E., Products of cocycles and extensions of mappings, Ann. of Math. (2), 48, 290-320, (1947) · Zbl 0030.41602
[12] Steenrod, N. E., Cohomology operations, (1962), Princeton University Press, Princeton, N.J. · Zbl 0102.38104
[13] Thomas, E., The suspension of the generalised pontrjagin cohomology operations, Pacific J. Math., 897-911, (1959) · Zbl 0121.39603
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.