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Differential calculus on almost commutative algebras and applications to the quantum hyperplane
Archivum mathematicum, Tome 41 (2005) no. 4, pp. 359-377 Cet article a éte moissonné depuis la source Czech Digital Mathematics Library

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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.
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.
Classification : 16E45, 16W35, 16W50, 58C50, 81R60
Keywords: noncommutative geometry; almost commutative algebra; linear connections; quantum hyperplane 
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Ciupală, Cătălin. Differential calculus on almost commutative algebras and applications to the quantum hyperplane. Archivum mathematicum, Tome 41 (2005) no. 4, pp. 359-377. http://geodesic.mathdoc.fr/item/ARM_2005_41_4_a1/

[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. | MR | Zbl

[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. | MR | Zbl

[3] Ciupală C.: Linear connections on almost commutative algebras. Acta Math. Univ. Comenian. 72, 2 (2003), 197–207. | MR | Zbl

[4] Ciupală C.: $\rho $-Differential calculi and linear connections on matrix algebra. Int. J. Geom. Methods Mod. Phys. 1 (2004), 847–863. | MR | Zbl

[5] Ciupală C.: Fields and forms on $\rho $-algebras. Proc. Indian Acad. Sci. Math. Sciences 112 (2005), 57–65. | MR | Zbl

[6] Connes A.: Non-commutative geometry. Academic Press, 1994.

[7] Dubois-Violette M.: Lectures on graded differential algebras and noncommutative geometry. Vienne, Preprint, E.S.I. 842 (2000). | MR | Zbl

[8] Dubois-Violette M., Michor P. W.: Connections on central bimodules. J. Geom. Phys. 20(1996), 218–232. | MR | Zbl

[9] Jadczyk A., Kastler D.: Graded Lie-Cartan pairs II. The fermionic differential calculus. Ann. Physics 179 (1987), 169–200. | MR | Zbl

[10] Kastler D.: Cyclic cohomology within the differential envelope. Hermann, Paris 1988. | MR | Zbl

[11] Lychagin V.: Colour calculus and colour quantizations. Acta Appl. Math. 41 (1995), 193–226. | MR | Zbl

[12] Mourad J.: Linear connections in non-commutative geometry. Classical Quantum Gravity 12 (1995), 965–974. | MR