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 ![]() ![]() |
[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, ![]() |
Quantum Grassmannian. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Quantum_Grassmannian&oldid=50165