nLab quantum flag manifold

Idea

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).

Literature

• 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

category: algebra