Abstract
Under the assumptions thatq is not a root of unity and that the differentialsdu i j of the matrix entries span the left module of first order forms, we classify bicovariant differential calculi on quantum groupsA n−1 ,B n ,C n andD n . We prove that apart one dimensional differential calculi and from finitely many values ofq, there are precisely2n such calculi on the quantum groupA n−1 =SL q (n) forn≧3. All these calculi have the dimensionn 2. For the quantum groupsB n ,C n andD n we show that except for finitely manyq there exist precisely twoN 2-dimensional bicovariant calculi forN≧3, whereN=2n+1 forB n andN=2n forC n ,D n . The structure of these calculi is explicitly described and the corresponding ad-invariant right ideals of ker ε are determined. In the limitq→1 two of the 2n calculi forA n−1 and one of the two calculi forB n ,C n andD n contain the ordinary classical differential calculus on the corresponding Lie group as a quotient.
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[BR] Barut, A.O., Rączka, R.: Theory of group representations and applications. PWN, Warsaw, 1977
[B] Bernard, D.: Quantum Lie algebras and differential calculus on quantum groups. Progr. Theor. Phys. Suppl.102, 49–66 (1990)
[BW] Birman, J., Wenzl, H.: Braids, link polynomials and a new algebra. Trans. Am. Math. Soc.313, 249–273 (1989)
[BM] Brzeziński, T., Majid, S.: A class of bicovariant differential calculi on Hopf algebras. Lett. Math. Phys.26, 67–78 (1992)
[CSWW] 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] Connes, A.: Non-commutative differential geometry. Publ. Math. IHES62, 44–144 (1986)
[D] Drinfeld, V.G.: Quantum groups. In: Proceedings ICM 1986, Providence, RI: Am. Math. Soc., 1987, pp. 798–820
[FRT] Faddeev, L.D., Reshetikhin, N.Yu., Takhtajan, L.A.: Quantization of Lie groups and Lie algebras, Leningrad Math. J.1, 193–225 (1990)
[H] Hammermesh, M.: Group theory and its application to physical problems. Reading, MA: Addison-Wesley, 1992
[J] 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)
[M] Manin, Yu.I.: Quantum groups and non-commutative geometry. Publications du C. R. M. 1561, Univ. of Montreal, 1988
[MH] Müller-Hoissen, F.: Differential calculi on the quantum groupGL p,q (2). J. Phys. A. Math. Gen.25, 1703–1734 (1992)
[PW] Parshall, B., Wang, I.: Quantum linear groups. Memoirs Am. Math. Soc.439, Providence, RI, 1991
[Re] Reshetikhin, N.Yu.: Quantized universal enveloping algebras, the Yang-Baxter equation and invariants of links I. Preprint LOMI E-4-87, Leningrad, 1988
[R1] Rosso, M.: Algèbres enveloppantes quantifiées, groupes quantiques compacts de matrices et calcul différentiel non commutatif. Duke Math. J.61, 11–40 (1990)
[R2] 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)
[SWZ] Schirrmacher, A., Wess, J., Zumino, B.: The two-parameter deformation ofGL(2), its differential calculus and Lie algebra. Z. Phys. C49, 317–324 (1991)
[SS] Schmüdgen, K., Schüler, A.: Covariant differential calculi on quantum spaces and on quantum groups. C.R. Acad. Sci. Paris316, 1155–1160 (1993).
[St] Stachura, P.: Bicovariant differential calculi onS μ U(2). Lett. Math. Phys.25, 175–188 (1992)
[Su] Sudberry, A.: Canonical differential calculus on quantum linear groups and supergroups. Phys. Lett. B284, 61–65 (1992)
[WZ] Wess, J., Zumino, B.: Covariant differential calculus on the quantum hyperplane. Nuclear Phys. B (Proc. Suppl.)18, 302–312 (1990)
[Wo1] Woronowicz, S.L.: TwistedSU(2) group. An example of noncommutative differential calculus. Publ. RIMS Kyoto Univ.23, 117–181 (1987)
[Wo2] Woronowicz, S.L.: Differential calculus on compact matrix pseudogroups (quantum groups). Commun. Math. Phys.122, 125–170 (1989)
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Schmüdgen, K., Schüler, A. Classification of bicovariant differential calculi on quantum groups of type A, B, C and D. Commun.Math. Phys. 167, 635–670 (1995). https://doi.org/10.1007/BF02101539
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DOI: https://doi.org/10.1007/BF02101539