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
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.
A. Joseph, Quantum groups and their primitive ideals, Springer 1995.
Quantum multilinear algebra, Mitsuyasu Hashimoto and Takahiro Hayashi Source: Tohoku Math. J. (2) Volume 44, Number 4 (1992), 471-521, euclid