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Le langage des espaces et des groupes quantiques. (The language of quantum spaces and quantum groups). (French) Zbl 0783.17007
A well known difficulty in non-commutative geometry is the following one: On a non-commutative algebra, there does not seem to exist a canonical choice of a differential structure (calculus of differential forms) coinciding in the special case of commutative algebras with the classical ones (given in a natural way by the underlying space). The author takes the point of view that, at least in the context of quantum groups and quantum spaces, the proper objects of a differential geometric study are not the usual quantum groups and spaces but graded differential algebras built over them. [Compare also Yu. I. Manin, Notes on quantum groups and quantum de Rham complexes, Teor. Mat. Fiz. 92, No. 3, 425-450 (1992).] The category of quantum spaces is defined as the opposite category to the category of graded differential algebras generated by the elements of degree zero. The main concern of the paper is to introduce the typical notions of the theory of quantum groups in this category.
After recalling some well-known facts about graded differential algebras the category of quantum spaces is defined. There are quantum subspaces, products of quantum spaces, quantum spaces of finite type. An important special case of a quantum space is a quantum cone (a bigraded differential algebra, the second gradation stemming from a gradation of the zeroth order (with respect to the first gradation) component). Then the notion of quantum monoid is introduced, which generalizes the notion of bialgebra. A special case are matrix quantum monoids. There exists the quantum monoid of endomorphisms of a quantum cone which is uniquely determined by some universality property. Quantum groups are defined as quantum monoids equipped with an antipode, which again means a generalization of the usual notion of antipode. For every matrix quantum monoid, there is an associated quantum group, which is a matrix quantum monoid under some additional assumption (the quantum monoid is Cramer). This last construction is a generalization of an analogous construction of Yu. I. Manin [Quantum groups and non-commutative geometry (Montreal, 1988; Zbl 0724.17006)], which itself is a generalization of the usual algebraic procedure of associating to a semigroup the group of its invertible elements.

17B37 Quantum groups (quantized enveloping algebras) and related deformations
81R50 Quantum groups and related algebraic methods applied to problems in quantum theory
46L85 Noncommutative topology
46L87 Noncommutative differential geometry
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[1] [Ab] Abe, E.: Hopf algebras. Cambridge Tracts in Math.74, Cambridge: Cambridge Univ. Press, 1980
[2] [Ao1] Aomoto, K.:q-analogue of the de Rham cohomology associated with Jackson integrals. I. Proc. Jap. Acad. A66, 161–164 (1990) · Zbl 0718.33011
[3] [Ao2] Aomoto, K.:q-analogue of the de Rham cohomology associated with Jackson integrals. II. Proc. Jap. Acad. A66, 240–244 (1990) · Zbl 0718.33012
[4] [Ao3] Aomoto, K.: Finiteness of a cohomology associated with certain Jackson integrals. Tôhoku Math. J.43, 75–101 (1991) · Zbl 0769.33016
[5] [AST] Artin, M., Schelter, W., Tate, J.: Quantum deformations ofGL n . Commun. Pure Appl. Math.44, 879–895 (1991) · Zbl 0753.17015
[6] [Be] Bergman, G.M.: The diamond lemma for ring theory. Adv. Math.29, 178–218 (1978) · Zbl 0377.16013
[7] [Bd] Bernard, D.: Quantum Lie algebras and differential calculus on quantum groups. Preprint 1990
[8] [Bo] Bourbaki, N.: Élements de Mathématique. Algèbre, chapitres 1 à 3, Paris: Hermann 1970
[9] [Br] Breziński, T.: Exterior bialgebras. Preprint 1991
[10] [BDR] Breziński T., Dabrowsky, H., Rembieliński, J.: On the quantum differential calculus and the quantum holomorphicity. J. Math. Phys.33, 1, 19–24 (1992) · Zbl 0753.17017
[11] [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) · Zbl 0743.17015
[12] [Ca] Cartier, P.: Calcul différentiel non commutatif, Exposés à l’E.N.S. (1989)
[13] [Co] Connes, A.: Non-commutative differential geometry. Publications Mathématiques de l’I.H.E.S.62, 41–144 (1985) · Zbl 0592.46056
[14] [DM] Deligne, P., Milne, J.S.: Tannakian categories. In: Hodge cycles, motives, and Shimura varieties. Lect. Notes in Math.900, Berlin, Heidelberg, New York: Springer 1982, pp. 101–228
[15] [Dr] Drinfel’d, V.G.: Quantum groups. In: Proceedings of the International Congress of Mathematicians 1986, Berkeley, Providence, RI: AMS, 1987, pp. 798–820
[16] [DV1] Dubois-Violette, M.: Dérivations et calcul différentiel non commutatif. C.R.A.S.307, 403–408 (1988) · Zbl 0661.17012
[17] [DV2] Dubois-Violette, M.: Noncommutative differential geometry, quantum mechanics and gauge theory. Preprint, L.P.T.H.E.-Orsay 90/31 (1990)
[18] [DKM1] Dubois-Violette, M., Kerner, R., Madore, J.: Noncommutative differential geometry of matrix algebras. J. Math. Phys.31, 2, 316–322 (1990) · Zbl 0704.53081
[19] [DKM2] Dubois-Violette M., Kerner, R., Madore, J.: Noncommutative differential geometry and new models of gauge theory. J. Math. Phys.31, 2, 323–330 (1990) · Zbl 0704.53082
[20] [FT] Feng, P., Tsygan, B.: Hochschild and cyclic homology of quantum groups. Commun. Math. Phys.140, 481–521 (1991) · Zbl 0743.17020
[21] [EGA IV4] Grothendieck, A.: Eléments de géométrie algébrique IV. Publications Mathématiques de l’I.H.E.S.32, (1967)
[22] [Gr] Grothendieck, A.: On the de Rham cohomology of algebraic varieties. Publications Mathématiques de l’I.H.E.S.29, 95–103 (1966) · Zbl 0145.17602
[23] [GRR] Gurevich, D., Radul, A., Rubtsov, V.: Non-commutative differential geometry and Yang-Baxter equation. Preprint 1990
[24] [HW] Hibi, T., Wakayama, M.: Aq-analogue of Capelli’s identity forGL(2). Preprint, Hokkaido University 1991
[25] [J1] Jimbo, M.: Aq-difference analogue ofU(g) and the Yang-Baxter equation. Lett. Math. Phys.10, 63–69 (1985) · Zbl 0587.17004
[26] [J2] Jimbo, M.: QuantumR matrix for the generalized Toda system Commun. Math. Phys.102, 537–547 (1986) · Zbl 0604.58013
[27] [J3] Jimbo, M.: Aq-analogue ofU(gl(N+1)), Hecke algebras and the Yang-Baxter equation. Lett. Math. Phys.11, 247–252 (1986) · Zbl 0602.17005
[28] [Ju] Jurčo, B.: Differential calculus on quantized simple Lie groups. Lett. Math. Phys.22, 177–186 (1991) · Zbl 0753.17020
[29] [Ka] Karoubi, M.: Homologie cyclique etK-théorie. Asterisque149, (1987)
[30] [Kas] Kassel, C.: Cyclic homology of differential operators, the Virasoro algebra and aq-analogue. Preprint 1991
[31] [Mal1] Maltsiniotis G.: Groupes quantiques et structures différentielles. C.R.A.S.311 Sér. I, 831–834 (1990) · Zbl 0728.17010
[32] [Mal2] Maltsiniotis, G.: Calcul différentiel sur le groupe linéaire quantique. Preprint 1990
[33] [Mal3] Maltsiniotis, G.: Groupoïdes quantiques. C.R.A.S.314, Sér. I, 249–252 (1992) · Zbl 0767.17015
[34] [Man1] Manin, Yu.I.: Some remarks on Koszul algebras and quantum groups. Ann. Inst. Fourier37, 4, 191–205 (1987) · Zbl 0625.58040
[35] [Man2] Manin, Yu.I.: Quantum groups and non-commutative geometry. Centre de Recherches Mathématiques de l’Université de Montréal 1988
[36] [Man3] Manin, Yu.I.: Multiparametric quantum deformation of the general linear supergroup. Commun. Math. Phys.123, 163–175 (1989) · Zbl 0673.16004
[37] [Man4] Manin, Yu.I.: Notes on quantum groups and quantum de Rham complexes. Preprint, MPI/91-60 (1991)
[38] [MNW1] Masuda, T., Nakagami, Y., Watanabe, J.: Noncommutative differential geometry on the quantumSU(2), I: An algebraic viewpointK-Theory4, 157–180 (1990) · Zbl 0719.46042
[39] [MNW2] Masuda, T., Nakagami, Y., Watanabe, J.: Noncommutative differential geometry on the quantum two sphere of Podles.I: An algebraic viewpoint.K-Theory5, 151–175 (1991) · Zbl 0763.46059
[40] [M-H] Müller-Hoissen, F.: Differential calculi on the quantum groupGL p,q (2). Preprint, GOET-TP 55/91 (1991)
[41] [NUW] Noumi, M., Umeda, T., Wakayama, M.: A quantum analogue of the Capelli identity and an elementary differential calculus onGL q (n). Preprint, University of Tokyo 1991
[42] [Po] Podles, P.: Differential calculus on quantum spheres. Lett Math. Phys.18, 107–119 (1989) · Zbl 0702.53073
[43] [RTF] Reshetikhin, N.Yu., Takhtadzhyan, L.A., Fadeev, L.D.: Quantization of Lie groups and Lie algebras. Len. Math. J.1, 1, 193–225 (1990) · Zbl 0715.17015
[44] [Ro] Rosso, M.: Algèbres enveloppantes quantifiées, groupes quantiques compacts de matrices et calcul différentiel non commutatif. Duke Math. J.61, 1, 11–40 (1990) · Zbl 0721.17013
[45] [Sch] Schirrmacher, A.: Remarks on the use ofR-matrices. Preprint 1991
[46] [SWZ] Schirrmacher, A., Wess, J., Zumino, B.: The two-parameter deformation ofGL(2), its differential calculus, and Lie algebra. Z. Phys. C–Particles and Fields49, 317–324 (1991)
[47] [SVZ] Schmidke, W.B., Vokos, S.P., Zumino, B.: Differential geometry of the quantum supergroupGL q (1/ 1). Z. Phys. C–Particles and Fields48, 249–255 (1990) · Zbl 0973.17504
[48] [Su] Sudbery, A.: Consistent multiparameter quantisation ofGL(n). J. Phys. A.: Math. Gen.23, L697-L704 (1990)
[49] [Ta] Takhtadzhyan, L.A.: Noncommutative homology of quantum tori. Funct. Anal. Appl.23, 2, 147–149 (1989) · Zbl 0708.19003
[50] [Tsy] Tsygan, B.: Notes on differential forms on quantum groups. Preprint 1991.
[51] [WZ] Wess, J., Zumino, B.: Covariant differential calculus on the quantum hyperplane. Preprint 1990 · Zbl 0957.46514
[52] [Wo1] Woronowicz, S.L.: TwistedSU(2) group. An example of a non-commutative differential calculus. Publ. RIMS23, 117–181 (1987) · Zbl 0676.46050
[53] [Wo2] Woronowicz, S.L.: Differential calculus on compact matrix pseudogroups (quantum groups). Commun. Math. Phys.122, 125–170 (1989) · Zbl 0751.58042
[54] [Za1] Zakrzewski, S.: A differential structure for quantum mechanics. J. Geom. Phys.2, 3, 135–145 (1985) · Zbl 0607.46041
[55] [Za2] Zakrzewski, S.: Quantum and classical pseudogroups. Part. II. Differential, and symplectic pseudogroups. Commun. Math. Phys.134, 371–395 (1990) · Zbl 0708.58031
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