HEISENBERG DOUBLE VERSUS DEFORMED DERIVATIVES
Abstract
Two approaches to the tangent space of a noncommutative space whose coordinate algebra is the enveloping algebra of a Lie algebra are known: the Heisenberg double construction and the approach via deformed derivatives, usually defined by procedures involving orderings among noncommutative coordinates or equivalently involving realizations via formal differential operators. In an earlier work, we rephrased the deformed derivative approach introducing certain smash product algebra twisting a semicompleted Weyl algebra. We show here that the Heisenberg double in the Lie algebra case, is isomorphic to that product in a nontrivial way, involving a datum ϕ parametrizing the orderings or realizations in other approaches. This way, we show that the two different formalisms, used by different communities, for introducing the noncommutative phase space for the Lie algebra type noncommutative spaces are mathematically equivalent.
References
- Phys. Rev. D 65, 084044 (2002), arXiv:hep-th/0105120 DOI: 10.1103/PhysRevD.65.084044. Crossref, Google Scholar
- SIGMA 6, 086 (2010), arXiv:1005.4429. Google Scholar
- Eur. Phys. J. C 36, 117 (2004), arXiv:hep-th/0310116 DOI: 10.1140/epjc/s2004-01887-0. Crossref, ADS, Google Scholar
- J. Algebra 309, 318 (2007), arXiv:math.RT/0604096. Crossref, ISI, Google Scholar
- J. Phys. A 39, 5189 (2006), arXiv:hep-th/0602036 DOI: 10.1088/0305-4470/39/18/030. Crossref, ADS, Google Scholar
- J. Algebra 202, 712 (1998), arXiv:q-alg/9701009 DOI: 10.1006/jabr.1997.7323. Crossref, ISI, Google Scholar
- Int. J. Math. 11, 523 (2000), arXiv:math.QA/9811174 DOI: 10.1016/S0129-167X(00)00026-X. ISI, Google Scholar
- Lett. Math. Phys. 66, 157 (2003), DOI: 10.1023/B:MATH.0000027508.00421.bf. Crossref, ISI, ADS, Google Scholar
- Eur. Phys. J. C 51, 229 (2007), arXiv:hep-th/0702215. Google Scholar
- Duke Math. J. 74, 763 (1994), DOI: 10.1215/S0012-7094-94-07428-0. Crossref, ISI, Google Scholar
- J. Lukierski and A. Nowicki, Heisenberg double description of κ-Poincaré algebra and κ-deformed phase space , arXiv:q-alg/9792003 . Google Scholar
- Int. J. Mod. Phys. A 42, 365204 (2009), arXiv:1004.4647. Google Scholar
- Eur. Phys. J. C 51, 229 (2007), arXiv:0705.2471 DOI: 10.1140/epjc/s10052-007-0285-8. Crossref, ISI, ADS, Google Scholar
- S. Meljanac and Z. Škoda, Leibniz rules for enveloping algebras , arXiv:0711.0149 . Google Scholar
- Eur. Phys. J. C 47, 531 (2006), arXiv:hep-th/0605133 DOI: 10.1140/epjc/s2006-02584-8. Crossref, ISI, ADS, Google Scholar
- S. Meljanac, D. Svrtan and Z. Škoda, Exponential formulas and Lie algebra type star products , arXiv:1006.0478 . Google Scholar
- Int. Math. Res. Notices 7, 143 (1992). Google Scholar
- Lett. Math. Phys. 92, 81 (2010), arXiv:0905.2215 DOI: 10.1007/s11005-010-0373-9. Crossref, ISI, ADS, Google Scholar
- Z. Škoda, Twisted exterior derivative for enveloping algebras , arXiv:0806.0978 . Google Scholar
