View source for Quantum Grassmannian

A subalgebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q120/q120020/q1200201.png" /> in the algebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q120/q120020/q1200202.png" /> of regular functions on the quantum group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q120/q120020/q1200203.png" /> (cf. [[Quantum groups|Quantum groups]]). <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q120/q120020/q1200204.png" /> is generated by quantum minors <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q120/q120020/q1200205.png" />, with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q120/q120020/q1200206.png" /> and with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q120/q120020/q1200207.png" /> the vector co-representation of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q120/q120020/q1200208.png" /> [[#References|[a1]]]. The <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q120/q120020/q1200209.png" />-minors satisfy quadratic relations, which turn into the Plücker relations (Young symmetries, cf. also [[Grassmann manifold|Grassmann manifold]]) when the deformation parameter <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q120/q120020/q12002010.png" /> is specialized to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q120/q120020/q12002011.png" />. Since classically the Grassmannian, as a complex submanifold in the projective space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q120/q120020/q12002012.png" />, is the common zero locus of the Plücker relations, one interprets <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q120/q120020/q12002013.png" /> as a quantization of the complex Poisson manifold <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q120/q120020/q12002014.png" /> (cf. [[Symplectic structure|Symplectic structure]]). The co-multiplication <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q120/q120020/q12002015.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q120/q120020/q12002016.png" /> induces a right co-action <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q120/q120020/q12002017.png" /> and so <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q120/q120020/q12002018.png" /> is a [[Quantum homogeneous space|quantum homogeneous space]].

A more general construction of (generalized) quantum flag manifolds exists for the group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q120/q120020/q12002019.png" /> [[#References|[a1]]], as well as for other simple complex Lie groups <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q120/q120020/q12002020.png" /> having quantum counterparts [[#References|[a2]]]. Another description was given in [[#References|[a3]]]. Both approaches [[#References|[a2]]], [[#References|[a3]]] also allow one to define quantum Schubert varieties.

Since <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q120/q120020/q12002021.png" /> is compact, the only holomorphic functions defined globally on it are the constants. But one can work instead with holomorphic coordinates <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q120/q120020/q12002022.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q120/q120020/q12002023.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q120/q120020/q12002024.png" />, on the big cell <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q120/q120020/q12002025.png" />, the unique Schubert cell of top dimension. The standard choice of coordinates is given via the Gauss decomposition of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q120/q120020/q12002026.png" />. For the algebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q120/q120020/q12002027.png" /> this means in fact a localization by allowing the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q120/q120020/q12002028.png" />-minor <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q120/q120020/q12002029.png" /> to be invertible. The generators <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q120/q120020/q12002030.png" /> of the quantum big cell <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q120/q120020/q12002031.png" /> satisfy the relations [[#References|[a4]]]

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The [[Symplectic manifold|symplectic manifold]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q120/q120020/q12002033.png" /> can be realized as an orbit of the dressing transformation of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q120/q120020/q12002034.png" /> acting on its dual [[Poisson Lie group|Poisson Lie group]]. The transformation can be also viewed as the right <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q120/q120020/q12002035.png" />-action on the manifold <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q120/q120020/q12002036.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q120/q120020/q12002037.png" /> unimodular positive matrices: <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q120/q120020/q12002038.png" />. The orbits are determined by sets of eigenvalues and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q120/q120020/q12002039.png" /> corresponds to a two-point set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q120/q120020/q12002040.png" /> with multiplicities <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q120/q120020/q12002041.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q120/q120020/q12002042.png" />, respectively. There exists a quantum analogue as a right co-action

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<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q120/q120020/q12002044.png" /> is endowed with a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q120/q120020/q12002045.png" />-involution and, correspondingly, one can turn <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q120/q120020/q12002046.png" /> into a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q120/q120020/q12002047.png" />-algebra by determining the commutation relations between <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q120/q120020/q12002048.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q120/q120020/q12002049.png" /> in dependence on the parameters <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q120/q120020/q12002050.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q120/q120020/q12002051.png" /> [[#References|[a4]]].

Similarly as for quantum spheres (cf. [[Quantum sphere|Quantum sphere]]), other types of quantum Grassmannians have been defined, distinguished by possessing classical points, i.e., one-dimensional representations <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q120/q120020/q12002052.png" /> [[#References|[a5]]].

====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  E. Taft,   J. Towber,   "Quantum deformations of flag schemes and Grassmann schemes I. A <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q120/q120020/q12002053.png" />-deformation for the shape algebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q120/q120020/q12002054.png" />"  ''J. Algebra'' , '''142'''  (1991)  pp. 1–36</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  Ya.S. Soibelman,   "On the quantum flag manifold"  ''Funct. Anal. Appl.'' , '''26'''  (1992)  pp. 225–227</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  V. Lakshmibai,   N. Reshetikhin,   "Quantum deformations of flag and Schubert schemes"  ''C.R. Acad. Sci. Paris'' , '''313'''  (1991)  pp. 121–126</TD></TR><TR><TD valign="top">[a4]</TD> <TD valign="top">  P. Šťovíček,   "Quantum Grassmann manifolds"  ''Comm. Math. Phys.'' , '''158'''  (1993)  pp. 135–153</TD></TR><TR><TD valign="top">[a5]</TD> <TD valign="top">  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, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q120/q120020/q12002055.png" />-Series and Related Topics'' , ''Fields Inst. Commun.'' , '''14''' , Amer. Math. Soc.  (1997)  pp. 167–177</TD></TR></table>