# Article

 Title: Differential calculus on almost commutative algebras and applications to the quantum hyperplane (English) Author: Ciupală, Cătălin Language: English Journal: Archivum Mathematicum ISSN: 0044-8753 (print) ISSN: 1212-5059 (online) Volume: 41 Issue: 4 Year: 2005 Pages: 359-377 Summary lang: English . Category: math . Summary: In this paper we introduce a new class of differential graded algebras named DG $\rho$-algebras and present Lie operations on this kind of algebras. We give two examples: the algebra of forms and the algebra of noncommutative differential forms of a $\rho$-algebra. Then we introduce linear connections on a $\rho$-bimodule $M$ over a $\rho$-algebra $A$ and extend these connections to the space of forms from $A$ to $M$. We apply these notions to the quantum hyperplane. (English) Keyword: noncommutative geometry Keyword: almost commutative algebra Keyword: linear connections Keyword: quantum hyperplane MSC: 16E45 MSC: 16W35 MSC: 16W50 MSC: 58C50 MSC: 81R60 idZBL: Zbl 1110.81111 idMR: MR2195490 . Date available: 2008-06-06T22:46:34Z Last updated: 2012-05-10 Stable URL: http://hdl.handle.net/10338.dmlcz/107966 . Reference: [1] Bongaarts P. J. M., Pijls H. G. J.: Almost commutative algebra and differential calculus on the quantum hyperplane.J. Math. Phys. 35 (2) 1994, 959–970. Zbl 0808.17011, MR 1257560 Reference: [2] Cap A., Kriegl A., Michor P. W., Vanžura J.: The Frölicher-Nijenhuis bracket in non commutative differential geometry.Acta Math. Univ. Comenian. 62 (1993), 17–49. Zbl 0830.58002, MR 1233839 Reference: [3] Ciupală C.: Linear connections on almost commutative algebras.Acta Math. Univ. Comenian. 72, 2 (2003), 197–207. Zbl 1087.81032, MR 2040264 Reference: [4] Ciupală C.: $\rho$-Differential calculi and linear connections on matrix algebra.Int. J. Geom. Methods Mod. Phys. 1 (2004), 847–863. Zbl 1063.58004, MR 2107309 Reference: [5] Ciupală C.: Fields and forms on $\rho$-algebras.Proc. Indian Acad. Sci. Math. Sciences 112 (2005), 57–65. Zbl 1086.58003, MR 2120599 Reference: [6] Connes A.: Non-commutative geometry.Academic Press, 1994. Reference: [7] Dubois-Violette M.: Lectures on graded differential algebras and noncommutative geometry.Vienne, Preprint, E.S.I. 842 (2000). Zbl 1038.58004, MR 1910544 Reference: [8] Dubois-Violette M., Michor P. W.: Connections on central bimodules.J. Geom. Phys. 20(1996), 218–232. Zbl 0867.53023, MR 1412695 Reference: [9] Jadczyk A., Kastler D.: Graded Lie-Cartan pairs II. The fermionic differential calculus.Ann. Physics 179 (1987), 169–200. Zbl 0637.17013, MR 0921314 Reference: [10] Kastler D.: Cyclic cohomology within the differential envelope.Hermann, Paris 1988. Zbl 0662.55001, MR 0932461 Reference: [11] Lychagin V.: Colour calculus and colour quantizations.Acta Appl. Math. 41 (1995), 193–226. Zbl 0846.18006, MR 1362127 Reference: [12] Mourad J.: Linear connections in non-commutative geometry.Classical Quantum Gravity 12 (1995), 965–974. MR 1330296 .

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