Quantum flag manifolds/varieties are the analogues of the usual flag manifiolds/varieties with the corresponding Lie groups replaced by quantum groups.
Quantum flag manifolds (or varieties) are certain quantum homogeneous spaces of duals of Drinfel’d-Jimbo quantum groups or their relatives, viewed as noncommutative quotients with respect to (co)action of the corresponding quantum Borel subgroup.
Usual flag varieties are projective varieties, hence they have a (commutative) graded coordinate ring. Thus most of the approaches to quantum flag varieties, define the quantum flag variety via its noncommutative graded algebra of functions (cf. the book by Joseph).
Yan S. Soĭbelʹman, О квантовом многообразии флагов, Функц. анализ и его прил. 26 (1992), no. 3, 90–92, pdf (Russian); translation On the quantum flag manifold, Funct. Anal. Appl. 26 (1992), no. 3, 225–227 doi
Jasper V. Stokman, Mathijs S. Dijkhuizen, Quantized flag manifolds and irreducible ∗-representations, Commun. Math. Phys. 203, 297–324 doi arXiv:math/9802086
V. A. Lunts, A. L. Rosenberg, Localization for quantum groups, Selecta Math. (N.S.) 5 (1999), no. 1, pp. 123–159 (doi).
Zoran Škoda, Localizations for construction of quantum coset spaces, in “Noncommutative geometry and Quantum groups”, W.Pusz, P.M. Hajac, eds. Banach Center Publications 61, pp. 265–298, Warszawa 2003, math.QA/0301090.
Ulrich Krähmer, Dirac operators on quantum flag manifolds, Lett. Math. Phys. 67/1 (2004) 49-59, MR2005b:58009, doi, math.QA/0305071
Anthony Joseph, Quantum groups and their primitive ideals, Springer 1995.
Earl Taft, Jacob Towber, Quantum deformation of flag schemes and Grassmann schemes q-deformation of the shape-algebra for GL(n), J. Algebra 142 (1991) 1–36 doi
Mitsuyasu Hashimoto, Takahiro Hayashi, Quantum multilinear algebra, Tohoku Math. J. (2) 44, n. 4 (1992), 471–521, euclid
Istvan Heckenberger, Stefan Kolb, The locally finite part of the dual coalgebra of quantized irreducible flag manifolds, Proc. London Math. Soc. (3), 89 (2004) 457–484 doi; De Rham complex for quantized irreducible flag manifolds, J. Algebra 305 (2006) 704–741 doi
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