Differential calculus on almost commutative algebras and applications to the quantum hyperplane

Cătălin Ciupală

Archivum Mathematicum (2005)

  • Volume: 041, Issue: 4, page 359-377
  • ISSN: 0044-8753

Abstract

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In this paper we introduce a new class of differential graded algebras named DG ρ -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  ρ -algebra. Then we introduce linear connections on a  ρ -bimodule M over a  ρ -algebra  A and extend these connections to the space of forms from A to M . We apply these notions to the quantum hyperplane.

How to cite

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Ciupală, Cătălin. "Differential calculus on almost commutative algebras and applications to the quantum hyperplane." Archivum Mathematicum 041.4 (2005): 359-377. <http://eudml.org/doc/249479>.

@article{Ciupală2005,
abstract = {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.},
author = {Ciupală, Cătălin},
journal = {Archivum Mathematicum},
keywords = {noncommutative geometry; almost commutative algebra; linear connections; quantum hyperplane; noncommutative geometry},
language = {eng},
number = {4},
pages = {359-377},
publisher = {Department of Mathematics, Faculty of Science of Masaryk University, Brno},
title = {Differential calculus on almost commutative algebras and applications to the quantum hyperplane},
url = {http://eudml.org/doc/249479},
volume = {041},
year = {2005},
}

TY - JOUR
AU - Ciupală, Cătălin
TI - Differential calculus on almost commutative algebras and applications to the quantum hyperplane
JO - Archivum Mathematicum
PY - 2005
PB - Department of Mathematics, Faculty of Science of Masaryk University, Brno
VL - 041
IS - 4
SP - 359
EP - 377
AB - 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.
LA - eng
KW - noncommutative geometry; almost commutative algebra; linear connections; quantum hyperplane; noncommutative geometry
UR - http://eudml.org/doc/249479
ER -

References

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  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. (1994) Zbl0808.17011MR1257560
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  5. Ciupală C., Fields and forms on ρ -algebras, Proc. Indian Acad. Sci. Math. Sciences 112 (2005), 57–65. Zbl1086.58003MR2120599
  6. Connes A., Non-commutative geometry, Academic Press, 1994. (1994) 
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  8. Dubois-Violette M., Michor P. W., Connections on central bimodules, J. Geom. Phys. 20(1996), 218–232. (1996) Zbl0867.53023MR1412695
  9. Jadczyk A., Kastler D., Graded Lie-Cartan pairs II. The fermionic differential calculus, Ann. Physics 179 (1987), 169–200. (1987) Zbl0637.17013MR0921314
  10. Kastler D., Cyclic cohomology within the differential envelope, Hermann, Paris 1988. (1988) Zbl0662.55001MR0932461
  11. Lychagin V., Colour calculus and colour quantizations, Acta Appl. Math. 41 (1995), 193–226. (1995) Zbl0846.18006MR1362127
  12. Mourad J., Linear connections in non-commutative geometry, Classical Quantum Gravity 12 (1995), 965–974. (1995) MR1330296

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