Quantum Grassmannian

A subalgebra in the algebra of regular functions on the quantum group (cf. Quantum groups). is generated by quantum minors , with and with the vector co-representation of [a1]. The -minors satisfy quadratic relations, which turn into the Plücker relations (Young symmetries, cf. also Grassmann manifold) when the deformation parameter is specialized to . Since classically the Grassmannian, as a complex submanifold in the projective space , is the common zero locus of the Plücker relations, one interprets as a quantization of the complex Poisson manifold (cf. Symplectic structure). The co-multiplication in induces a right co-action and so is a quantum homogeneous space.

A more general construction of (generalized) quantum flag manifolds exists for the group [a1], as well as for other simple complex Lie groups having quantum counterparts [a2]. Another description was given in [a3]. Both approaches [a2], [a3] also allow one to define quantum Schubert varieties.

Since is compact, the only holomorphic functions defined globally on it are the constants. But one can work instead with holomorphic coordinates , , , on the big cell , the unique Schubert cell of top dimension. The standard choice of coordinates is given via the Gauss decomposition of . For the algebra this means in fact a localization by allowing the -minor to be invertible. The generators of the quantum big cell satisfy the relations [a4]

The symplectic manifold can be realized as an orbit of the dressing transformation of acting on its dual Poisson Lie group. The transformation can be also viewed as the right -action on the manifold of unimodular positive matrices: . The orbits are determined by sets of eigenvalues and corresponds to a two-point set with multiplicities and , respectively. There exists a quantum analogue as a right co-action

is endowed with a -involution and, correspondingly, one can turn into a -algebra by determining the commutation relations between and in dependence on the parameters and [a4].

Similarly as for quantum spheres (cf. Quantum sphere), other types of quantum Grassmannians have been defined, distinguished by possessing classical points, i.e., one-dimensional representations [a5].

References

[a1]	E. Taft, J. Towber, "Quantum deformations of flag schemes and Grassmann schemes I. A -deformation for the shape algebra " J. Algebra , 142 (1991) pp. 1–36
[a2]	Ya.S. Soibelman, "On the quantum flag manifold" Funct. Anal. Appl. , 26 (1992) pp. 225–227
[a3]	V. Lakshmibai, N. Reshetikhin, "Quantum deformations of flag and Schubert schemes" C.R. Acad. Sci. Paris , 313 (1991) pp. 121–126
[a4]	P. Šťovíček, "Quantum Grassmann manifolds" Comm. Math. Phys. , 158 (1993) pp. 135–153
[a5]	M. Nuomi, M.S. Dijkhuizen, T. Sugitani, "Multivariable Askey–Wilson polynomials and quantum complex Grassmannians" M.E.H. Insmail (ed.) et al. (ed.) , Special Functions, -Series and Related Topics , Fields Inst. Commun. , 14 , Amer. Math. Soc. (1997) pp. 167–177