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Quantized Koszul complexes. (Complexes de Koszul quantiques.) (French) Zbl 0810.16010
The results of this work were partially announced by the author [in C. R. Acad. Sci., Paris, Sér. I 314, 977-982 (1992; Zbl 0760.16004)] and concern a generalization of the Koszul complex associated with symmetries satisfying the Yang-Baxter equation [D. I. Gurevich, Algebra Anal. 2, No. 4, 119-148 (1990; Zbl 0713.17010), V. V. Lyubashenko, Usp. Mat. Nauk 41, No. 5, 185-186 (1986; Zbl 0649.16008), Yu. I. Manin, “Quantum Groups and Non-Commutative Geometry (C.R.M. Univ. Montréal 1988; Zbl 0724.17006)] and especially with Hecke symmetries. In this work the author demonstrates that this complex is acyclic and next calculates its Hochschild homology. The results are applied to multiparametric affine spaces and “quantum” Lie algebras.

MSC:
16E40 (Co)homology of rings and associative algebras (e.g., Hochschild, cyclic, dihedral, etc.)
17B37 Quantum groups (quantized enveloping algebras) and related deformations
16W30 Hopf algebras (associative rings and algebras) (MSC2000)
17B56 Cohomology of Lie (super)algebras
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References:
[1] A. BEILINSON, V. GINZBURG, W. SOERGEL, Koszul duality patterns in representation theory, preprint. · Zbl 0864.17006
[2] M. VAN DEN BERGH, Non-commutative homology of some three-dimensional quantum spaces, preprint Université Louis Pasteur, Strasbourg. · Zbl 0814.16006
[3] N. BOURBAKI, Eléments de mathématique, Algèbre, Chapitre X : Algèbre homologique, Paris, Masson, 1980. · Zbl 0455.18010
[4] H. CARTAN, S. EILENBERG, Homological algebra, Princeton, Princeton University Press, 1956. · Zbl 0075.24305
[5] A. CONNES, Non-commutative differential geometry, Publ. I.H.E.S., 62 (1985), 41-144. · Zbl 0592.46056
[6] CHR. CUVIER, Homologie des algèbres de Leibniz, C.R. Acad. Sci. Paris, 313 (1991), 569-572. · Zbl 0751.17018
[7] P. FENG, B. TSYGAN, Hochschild and cyclic homology of quantum groups, Comm. Math. Phys., 140 (1991) 481-521. · Zbl 0743.17020
[8] J.A. GUCCIONE, J.J. GUCCIONE, Hochschild and cyclic homology of Ore’s extensions, preprint Université de Buenos Aires. · Zbl 0886.16009
[9] D.I. GUREVICH, Algebraic aspects of quantum Yang-Baxter equation, Algebra i Analiz, 2 (1990), 119-148 ; traduction anglaise : Leningrad Math. J., 2 (1991), 801-829. · Zbl 0728.17012
[10] G. HOCHSCHILD, B. KOSTANT, A. ROSENBERG, Differential forms on regular affine algebras, Trans. A.M.S., 102 (1962), 383-408. · Zbl 0102.27701
[11] N. JACOBSON, Lie algebras, New York, Dover Publication Inc., 1979. · Zbl 0121.27504
[12] A. JOYAL, R. STREET, The geometry of tensor calculus (I), Adv. in Math., 88 (1991), 55-112. · Zbl 0738.18005
[13] CHR. KASSEL, L’homologie cyclique des algèbres enveloppantes, Invent. Math., 91 (1989), 221-251. · Zbl 0653.17007
[14] CHR. KASSEL, Cyclic homology of differential operators, the Virasoro Algebra and a q-Analogue, Commun. Math. Phys., 146 (1992), 343-356. · Zbl 0761.17020
[15] CHR. KASSEL, Quantum groups, (en préparation).
[16] CHR. KASSEL, V. TURAEV, Double construction for monoidal categories, preprint I.R.M.A., Strasbourg, (1992).
[17] J.-L. LODAY, D. QUILLEN, Cyclic homology and the Lie algebra homology of matrices, Comment. Math. Helvetici, 59 (1984), 565-591. · Zbl 0565.17006
[18] V.V. LYUBASHENKO, Hopf algebra and vector symmetries, Uspekhi Mat. Nauk, 41,5 (1986), 185-186 ; traduction anglaise : Russian Math. Surveys, 41,5 (1986), 153-154. · Zbl 0649.16008
[19] YU.I. MANIN, Quantum groups and non-commutative geometry, C.R.M., Université de Montréal, 1988. · Zbl 0724.17006
[20] N. YU RESHETIKHIN, V.G. TURAEV, Ribbon graphs and their invariants derived from quantum groups, Commun. Math. Phys., 127 (1990), 1-26. · Zbl 0768.57003
[21] M. ROSSO, Algèbres enveloppantes quantifiées, groupes quantiques compacts de matrices et calcul différentiel non commutatif., Duke Math. J., 61 (1990), 1-26. · Zbl 0721.17013
[22] A. SOLOTAR, M. REDONDO, Hochschild homology of q-differential operators, (preprint).
[23] R. SRIDHARAN, Filtered algebras and representations, Trans. Am. Math. Soc., 100 (1961), 530-550. · Zbl 0099.02301
[24] L.A. TAKHTADJIAN, Noncommutative homology of quantum tori, Funkt. Anal. Pril., 24,2 (1989), 75-76 ; traduction anglaise : Funct. Anal. Appl., 23 (1989), 147-149. · Zbl 0708.19003
[25] M. WAMBST, Complexes de Koszul quantiques et homologie cyclique, C.R. Acad. Sci. Paris, 314 (1992), 977-982. · Zbl 0760.16004
[26] S.L. WORONOWICZ, Differential calculus on compact matrix pseudo-groups, Commun. Math. Phys., 122 (1989), 125-170. · Zbl 0751.58042
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