Abstract
Suppose thatq is not a root of unity. We classify all bicovariant differential calculi of dimension greater than one on the quantum groupsGL q (N),O q (N) andSp q (N) for which the differentials du i j of the matrix entriesu i j generate the left module of first order forms. Our first classification theorem asserts that there are precisely two one-parameter families of such calculi onGL q (N) forN≧3. In the limitq→1 only two of these calculi give the ordinary differential calculus onGL(N). Our second main theorem states that apart from finitely manyq there exist precisely two differential calculi with these properties onO q (N) andSp q (N) forN≧4. This strengthens the corresponding result proved in our previous paper [SS2]. There are four such calculi onO q (3). We introduce two new 4-dimensional bicovariant differential calculi onO q (3).
Similar content being viewed by others
References
[BM] Brzeziński, T., Majid, S.: Quantum group gauge theory on quantum spaces. Commun. Math. Phys.157, 591–638 (1993)
[CSSW] Carow-Watamura, U., Schlieker, M., Watamura, S., Weich, W.: Bicovariant differential calculus on quantum groupsSU q (N) andSO q (N). Commun. Math. Phys.142, 605–641 (1991)
[C] Castellani, L.: Gauge theories of quantum groups. Phys. Lett.B292, 93–98 (1992)
[D] Drinfeld, V.G.: Quantum groups. In: Proceedings ICM 1986, pp. 798–820, Providence, RI: Am. Math. Soc., 1987
[DK] Dijkhuizen, M.S., Koornwinder, T.W.: Quantum homogeneous spaces, duality and quantum 2-spheres. Preprint, Amsterdam, 1993
[FRT] Faddeev, L.D., Reshetikhin, N.Yu., Takhatajan, L.A.: Quantization of Lie groups and Lie algebras. Algebra and Analysis1, 178–206 (1987)
[Hm] Hammermesh, M.: Group theory and its application to physical problems. Reading, MA: Addison-Wesley, 1992
[H] Hayashi, T.: Quantum groups and quantum determinants. J. Algebra152, 146–165 (1992)
[J] Jimbo, M.: Aq-difference analogue ofU(g) and the Yang-Baxter equation. Lett. Math. Phys.10, 63–65 (1985)
[Ju] Jurčo, B.: Differential calculus on quantized simple Lie groups. Lett. Math. Phys.22, 177–186 (1991)
[L] Lusztig, G.: Quantum deformations of certain simple modules over, enveloping algebras. Adv. Math.70, 237–249 (1988)
[Mj] Majid, S.: Quasitringular Hopf algebras and Yang-Baxter, equations. Int. J. Mod. Phys.A5, 1–91 (1990)
[M1] Maltsiniotis, G.: Calcul différential sur le groupe lineáire quantique. Preprint, ENS, Paris, 1990
[M1] Manin, Yu. I.: Quantum groups and non-commutative geometry. Publications du C.R.M. 1561, Univ. of Montreal, (1988)
[M2] Manin, Yu. I.: Notes on quantum groups and quantum de Rham complexes. Preprint, Max-Planck-Institut, Bonn, (1991)
[MH] Müller-Hoissen, F.: Differential calculi on the quantum groupGL p,q (2). J. Phys. A. Math. Gen.25, 1703–1734 (1992)
[S] Sudberry, A.: Canonical differential calculus on quantum linear groups and supergroups. Phys. Lett. B284, 61–65 (1992)
[PW] Parshall, B., Wang, I.: Quantum linear groups. Memoirs Am. Math. Soc.439, Providence, RI (1991)
[R] Rosso, M.: Finite dimensional representations of the quantum analog of the enveloping algebra of a complex simple Lie algebra. Commun. Math. Phys.117, 581–593 (1988)
[SS1] Schmüdgen, K., Schüler, A.: Covariant differential calculi on quantum spaces and on quantum groups. C.R. Acad. Sci. Paris316, 1155–1160 (1993)
[SS2] Schmüdgen, K., Schüler, A.: Classification of bicovariant calculi on quantum groups of typeA,B,C andD. Commun. Math. Phys.167, 635–670 (1995)
[SWZ] Schupp, P., Watts, P., Zumino, B.: Differential Geometry on Linear Quantum Groups. Lett. Math. Phys.25, 139–147 (1992)
[T] Takeuchi, M.: Matric bialgebras and quantum groups. Israel J. Math.72, 232–251 (1990)
[W] Woronowicz, S.L.: Differential calculus on compact matrix pseudogroups (quantum groups). Commun Math. Phys.122, 125–170 (1989)
Author information
Authors and Affiliations
Additional information
Communicated by M. Jimbo
Rights and permissions
About this article
Cite this article
Schmüdgen, K., Schüler, A. Classification of bicovariant differential calculi on quantum groups. Commun.Math. Phys. 170, 315–335 (1995). https://doi.org/10.1007/BF02108331
Received:
Revised:
Issue Date:
DOI: https://doi.org/10.1007/BF02108331