| 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 |
| . |
