Taft, Earl; Towber, Jacob Quantum deformation of flag schemes and Grassmann schemes. I: A \(q\)- deformation of the shape-algebra for \(GL(n)\). (English) Zbl 0739.17007 J. Algebra 142, No. 1, 1-36 (1991). The authors construct \(q\)-deformations of the affine Grassmann and flag varieties over the quantum general and special linear groups. Here \(q\)- deformations mean deformations of the coordinate rings of the appropriate varieties. This is done by deforming the relations given by the second author [in J. Algebra 47, 80-104 (1977; Zbl 0358.15033) and ibid. 61, 414-462 (1979; Zbl 0437.14030)]. Reviewer: H.H.Andersen (Aarhus) Cited in 4 ReviewsCited in 45 Documents MSC: 17B37 Quantum groups (quantized enveloping algebras) and related deformations 14M17 Homogeneous spaces and generalizations Keywords:Grassmann variety; \(q\)-deformations; flag varieties Citations:Zbl 0358.15033; Zbl 0437.14030 PDFBibTeX XMLCite \textit{E. Taft} and \textit{J. Towber}, J. Algebra 142, No. 1, 1--36 (1991; Zbl 0739.17007) Full Text: DOI References: [1] Abe, E., Hopf Algebras (1977), Cambridge Univ. Press: Cambridge Univ. Press London/New York [2] Bergman, G. M., The diamond lemma for ring theory, Adv. in Math., 29, 178-218 (1978) · Zbl 0326.16019 [3] Drin’feld, V. G., Soviet Math. Dokl., 32, 254-288 (1985) [4] Drin’feld, V. G., Quantum Groups, (Proc. ICM. Proc. ICM, Berkeley (1986)) [5] Faddeev, L. D.; Reshetikhin, N. Yu; Takhtajan, D. A., Quantized Lie groups and Lie algebras, LOMI — preprint E-14-87 (1987), Leningrad [6] Faddeev, L. D.; Takhtajan, D. A., Liouville model on the lattice, (Lecture Notes in Physics, Vol. 246 (1986), Springer-Verlag: Springer-Verlag New York/Berlin), 166-179 [8] Jimbo, M., A \(q\)-analog of \(U( gl (N + 1))\), Hecke algebra, and the Yang-Baxter equations, Lett. Math. Phys., 11, 247-252 (1986) · Zbl 0602.17005 [9] Jimbo, M., Quantum matrix for the generalized Toda system, Comm. Math. Phys., 102, 537-548 (1986) · Zbl 0604.58013 [10] Kulish, P. P.; Reshetikhin, N. Yu, Quantum linear problem for the Sine-Gordon equation and higher representations, J. Soviet Math., 23, No.4, 2435-2441 (1983), (101-11) (translated from 1981 paper) · Zbl 0545.35082 [11] Lusztig, G., Quantum deformations of certain simple modules over enveloping algebras, Adv. in Math., 70, 237-249 (1988) · Zbl 0651.17007 [12] Manin, Yu I., Some remarks on Koszul algebras and quantum groups, Ann. Inst. Fourier Grenoble, 37, 191-205 (1987) · Zbl 0625.58040 [13] Manin, Yu I., Quantum Groups and Non-Commutative Geometry (1988), Publications du C.R.M., Université de Montréal · Zbl 0724.17006 [14] Rosso, M., Représentations irréductibles de dimension finie du \(q\)-analogue de l’algèbre enveloppante d’une algèbre de Lie simple, C.R. Acad. Sci. Paris Sér. I, 305, 587-590 (1987) · Zbl 0624.17005 [15] Sklyanin, E. K., Some algebraic structures connected with the Yang-Baxter equation, Functional Anal. Appl., 10, 263-270 (1982), (translated from paper submitted in 1981) · Zbl 0513.58028 [16] Towber, J., Two new functions from modules to algebras, J. Algebra, 47, 80-104 (1977) · Zbl 0358.15033 [17] Towber, J., Young symmetry, the flag manifold, and representations of \(GL (n)\), J. Algebra, 61, 414-462 (1979) · Zbl 0437.14030 [18] Woronowicz, S. L., Compact matrix pseudogroups, Comm. Math. Phys., 3, 613-665 (1987) · Zbl 0627.58034 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.