Beilinson, A. A.; Lusztig, George; MacPherson, Robert A geometric setting for the quantum deformation of \(\mathrm{GL}_n\). (English) Zbl 0713.17012 Duke Math. J. 61, No. 2, 655-677 (1990). For the quantum deformation of \(\mathrm{GL}_n\) a realization is given not in terms of generators and relations which, according to Drinfel’d, “do not give much to heart or imagination” but in geometric terms of relative positions of pairs of flags in an infinite dimensional space. This infinite case is approximated by finite-dimensional ones via a nontrivial limiting procedure. Reviewer: Dimitry Leites (Stockholm) Cited in 37 ReviewsCited in 149 Documents MSC: 17B37 Quantum groups (quantized enveloping algebras) and related deformations 14C17 Intersection theory, characteristic classes, intersection multiplicities in algebraic geometry 16T20 Ring-theoretic aspects of quantum groups 81R50 Quantum groups and related algebraic methods applied to problems in quantum theory Keywords:quantum groups; infinite dimensional flags; intersection cohomology; quantum deformation PDFBibTeX XMLCite \textit{A. A. Beilinson} et al., Duke Math. J. 61, No. 2, 655--677 (1990; Zbl 0713.17012) Full Text: DOI References: [1] R. Dipper and S. Donkin, Quantum \(GL_n\) , [2] V. G. Drinfeld, Quantum groups , Proceedings of the International Congress of Mathematicians, Vol. 1, 2 (Berkeley, Calif., 1986), Amer. Math. Soc., Providence, RI, 1987, pp. 798-820. · Zbl 0667.16003 [3] M. Jimbo, A \(q\)-analogue of \(U(\mathfrak g\mathfrak l(N+1))\), Hecke algebra, and the Yang-Baxter equation , Lett. Math. Phys. 11 (1986), no. 3, 247-252. · Zbl 0602.17005 · doi:10.1007/BF00400222 [4] D. Kazhdan and G. Lusztig, Representations of Coxeter groups and Hecke algebras , Invent. Math. 53 (1979), no. 2, 165-184. · Zbl 0499.20035 · doi:10.1007/BF01390031 [5] D. Kazhdan and G. Lusztig, Schubert varieties and Poincaré duality , Geometry of the Laplace operator (Proc. Sympos. Pure Math., Univ. Hawaii, Honolulu, Hawaii, 1979), Proc. Sympos. Pure Math., XXXVI, Amer. Math. Soc., Providence, R.I., 1980, pp. 185-203. · Zbl 0461.14015 [6] G. Lusztig, Finite-dimensional Hopf algebras arising from quantized universal enveloping algebra , J. Amer. Math. Soc. 3 (1990), no. 1, 257-296. JSTOR: · Zbl 0695.16006 · doi:10.2307/1990988 [7] G. Lusztig, Canonical bases arising from quantized enveloping algebras , J. Amer. Math. Soc. 3 (1990), no. 2, 447-498. JSTOR: · Zbl 0703.17008 · doi:10.2307/1990961 [8] C. M. Ringel, Hall algebras and quantum groups , preprint to appear in Invent. Math. · Zbl 0735.16009 · doi:10.1007/BF01231516 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.