×

zbMATH — the first resource for mathematics

Graded algebras with symmetries. (Algèbres graduées avec symétries.) (French) Zbl 1238.16008
Author’s summary: We define the notion of “graded algebra with symmetries”. This notion is a generalization of the extended differential forms. We prove that for a graded algebra with symmetries \(T\), we associate a subalgebra \(\Omega^*\) which generalizes the noncommutative differential forms. Using this algebra \(\Omega^*\), we can define the Hochschild and cyclic homologies, cup \(i\)-products and the Steenrod squares.

MSC:
16E45 Differential graded algebras and applications (associative algebraic aspects)
16E40 (Co)homology of rings and associative algebras (e.g., Hochschild, cyclic, dihedral, etc.)
55S10 Steenrod algebra
58A10 Differential forms in global analysis
18G55 Nonabelian homotopical algebra (MSC2010)
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] Battikh, N., Cup i-produits sur LES formes différentielles non commutatives et carrés de Steenrod, J. algebra, 313, 531-553, (2007) · Zbl 1117.55015
[2] Connes, A., Noncommutative differential geometry, Publ. math. inst. hautes etudes sci., 62, 257-360, (1985)
[3] Cuntz, J.; Quillen, D., Operators on noncommutative differential forms and cyclic homology, () · Zbl 0865.18009
[4] Dold, A.; Thom, R., Une généralisation de la notion dʼespace fibré. application aux produits symétriques infinis, C. R. math. acad. sci. Paris, 242, 1680-1682, (1956) · Zbl 0071.17301
[5] Karoubi, M., Homologie cyclique et K-théorie, Astérisque, 149, (1987) · Zbl 0648.18008
[6] Karoubi, M., Formes topologiques non commutatives, Ann. sci. ec. norm. super., 28, 477-492, (1995) · Zbl 0837.55004
[7] Karoubi, M., Formes différentielles non commutatives et cohomologie à coefficients arbitraires, Trans. amer. math. soc., 347, 4277-4299, (1995) · Zbl 0852.55009
[8] Loday, J.-L., Opérations sur lʼhomologie cyclique des algèbres commutatives, Invent. math., 96, 205-230, (1989) · Zbl 0686.18006
[9] J.L. Loday, La renaissance des opérades, Séminaire N. Bourbaki, 1994-1995, exp. n^{o} 792, pp. 47-74.
[10] Steenrod, N.E.; Epstein, D.B.A., Cohomology operations, Ann. of math. stud., (1962), Princeton University Press · Zbl 0521.55001
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.