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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]].
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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.
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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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A subalgebra $\mathcal{G} = \operatorname { Fun } _ { q } ( G ( k , n ) )$ in the algebra $\mathcal{A} = \operatorname { Fun } _ { q } ( \operatorname{SL} ( n , \mathbf{C} ) )$ of regular functions on the quantum group $\operatorname{SL} ( n , \mathbf{C} )$ (cf. [[Quantum groups|Quantum groups]]). $\mathcal{G}$ is generated by quantum minors $| T _ { i _ { 1 } , \ldots , i _ { k } } ^ { 1 , \ldots , k } | _ { q }$, with $1 \leq i _ { 1 } &lt; \ldots &lt; i _ { k } \leq n$ and with $T \in \operatorname { Mat } ( n ) \otimes \mathcal{A}$ the vector co-representation of $\mathcal{A}$ [[#References|[a1]]]. The $q$-minors satisfy quadratic relations, which turn into the Plücker relations (Young symmetries, cf. also [[Grassmann manifold|Grassmann manifold]]) when the deformation parameter $q$ is specialized to $1$. Since classically the Grassmannian, as a complex submanifold in the projective space $\mathbf{P} ( \wedge ^ { k } \mathbf{C} ^ { n } )$, is the common zero locus of the Plücker relations, one interprets $\mathcal{G}$ as a quantization of the complex Poisson manifold $G ( k , n )$ (cf. [[Symplectic structure|Symplectic structure]]). The co-multiplication $\Delta$ in $\mathcal{A}$ induces a right co-action $R = \Delta |_{\cal G} :\cal G \rightarrow G \otimes A$ and so $\mathcal{G}$ is a [[Quantum homogeneous space|quantum homogeneous space]].
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q120/q120020/q12002032.png" /></td> </tr></table>
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A more general construction of (generalized) quantum flag manifolds exists for the group $\operatorname{SL} ( n , \mathbf{C} )$ [[#References|[a1]]], as well as for other simple complex Lie groups $G$ 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.
  
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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Since $G ( k , n )$ is compact, the only holomorphic functions defined globally on it are the constants. But one can work instead with holomorphic coordinates $z _ { s t }$, $1 \leq s \leq k$, $1 \leq t \leq n - k$, on the big cell $C$, the unique Schubert cell of top dimension. The standard choice of coordinates is given via the Gauss decomposition of $T$. For the algebra $\mathcal{G}$ this means in fact a localization by allowing the $q$-minor $| T _ { 1, \dots, k } ^ { 1 , \ldots , k } | _ { q }$ to be invertible. The generators $z _ { s t }$ of the quantum big cell $\mathcal{C} = \operatorname { Fun } _ { q } ( C )$ satisfy the relations [[#References|[a4]]]
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q120/q120020/q12002043.png" /></td> </tr></table>
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<table class="eq" style="width:100%;"> <tr><td style="width:94%;text-align:center;" valign="top"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q120/q120020/q12002032.png"/></td> </tr></table>
  
<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]]].
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The [[Symplectic manifold|symplectic manifold]] $G ( k , n )$ can be realized as an orbit of the dressing transformation of $\operatorname{SU} ( n )$ acting on its dual [[Poisson Lie group|Poisson Lie group]]. The transformation can be also viewed as the right $\operatorname{SU} ( n )$-action on the manifold $M$ of $( n \times n )$ unimodular positive matrices: $( m , u ) \mapsto u ^ { * } m u$. The orbits are determined by sets of eigenvalues and $G ( k , n )$ corresponds to a two-point set $\{ \lambda _ { 1 } , \lambda _ { 2 } \}$ with multiplicities $k$ and $n - k$, respectively. There exists a quantum analogue as a right co-action
  
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]]].
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\begin{equation*} \operatorname { Fun } _ { q } ( M ) \rightarrow \operatorname { Fun } _ { q } ( M ) \bigotimes \operatorname { Fun } _ { q } ( \operatorname {SU} ( n ) ). \end{equation*}
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$\operatorname { Fun } _ { q } ( M )$ is endowed with a $*$-involution and, correspondingly, one can turn $\mathcal{C}$ into a $*$-algebra by determining the commutation relations between $z _ { s t }$ and $z ^ { * _{ u v}}$ in dependence on the parameters $\lambda _ { 1 }$ and $\lambda _ { 2 }$ [[#References|[a4]]].
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 +
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 $\operatorname {Fun}_q ( G ( k , n ) ) \rightarrow \mathbf C $ [[#References|[a5]]].
  
 
====References====
 
====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>
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<table><tr><td valign="top">[a1]</td> <td valign="top">  E. Taft,  J. Towber,  "Quantum deformations of flag schemes and Grassmann schemes I. A $q$-deformation for the shape algebra $\operatorname{GL} ( n )$"  ''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, $q$-Series and Related Topics'' , ''Fields Inst. Commun.'' , '''14''' , Amer. Math. Soc.  (1997)  pp. 167–177</td></tr></table>

Revision as of 16:56, 1 July 2020

A subalgebra $\mathcal{G} = \operatorname { Fun } _ { q } ( G ( k , n ) )$ in the algebra $\mathcal{A} = \operatorname { Fun } _ { q } ( \operatorname{SL} ( n , \mathbf{C} ) )$ of regular functions on the quantum group $\operatorname{SL} ( n , \mathbf{C} )$ (cf. Quantum groups). $\mathcal{G}$ is generated by quantum minors $| T _ { i _ { 1 } , \ldots , i _ { k } } ^ { 1 , \ldots , k } | _ { q }$, with $1 \leq i _ { 1 } < \ldots < i _ { k } \leq n$ and with $T \in \operatorname { Mat } ( n ) \otimes \mathcal{A}$ the vector co-representation of $\mathcal{A}$ [a1]. The $q$-minors satisfy quadratic relations, which turn into the Plücker relations (Young symmetries, cf. also Grassmann manifold) when the deformation parameter $q$ is specialized to $1$. Since classically the Grassmannian, as a complex submanifold in the projective space $\mathbf{P} ( \wedge ^ { k } \mathbf{C} ^ { n } )$, is the common zero locus of the Plücker relations, one interprets $\mathcal{G}$ as a quantization of the complex Poisson manifold $G ( k , n )$ (cf. Symplectic structure). The co-multiplication $\Delta$ in $\mathcal{A}$ induces a right co-action $R = \Delta |_{\cal G} :\cal G \rightarrow G \otimes A$ and so $\mathcal{G}$ is a quantum homogeneous space.

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

Since $G ( k , n )$ is compact, the only holomorphic functions defined globally on it are the constants. But one can work instead with holomorphic coordinates $z _ { s t }$, $1 \leq s \leq k$, $1 \leq t \leq n - k$, on the big cell $C$, the unique Schubert cell of top dimension. The standard choice of coordinates is given via the Gauss decomposition of $T$. For the algebra $\mathcal{G}$ this means in fact a localization by allowing the $q$-minor $| T _ { 1, \dots, k } ^ { 1 , \ldots , k } | _ { q }$ to be invertible. The generators $z _ { s t }$ of the quantum big cell $\mathcal{C} = \operatorname { Fun } _ { q } ( C )$ satisfy the relations [a4]

The symplectic manifold $G ( k , n )$ can be realized as an orbit of the dressing transformation of $\operatorname{SU} ( n )$ acting on its dual Poisson Lie group. The transformation can be also viewed as the right $\operatorname{SU} ( n )$-action on the manifold $M$ of $( n \times n )$ unimodular positive matrices: $( m , u ) \mapsto u ^ { * } m u$. The orbits are determined by sets of eigenvalues and $G ( k , n )$ corresponds to a two-point set $\{ \lambda _ { 1 } , \lambda _ { 2 } \}$ with multiplicities $k$ and $n - k$, respectively. There exists a quantum analogue as a right co-action

\begin{equation*} \operatorname { Fun } _ { q } ( M ) \rightarrow \operatorname { Fun } _ { q } ( M ) \bigotimes \operatorname { Fun } _ { q } ( \operatorname {SU} ( n ) ). \end{equation*}

$\operatorname { Fun } _ { q } ( M )$ is endowed with a $*$-involution and, correspondingly, one can turn $\mathcal{C}$ into a $*$-algebra by determining the commutation relations between $z _ { s t }$ and $z ^ { * _{ u v}}$ in dependence on the parameters $\lambda _ { 1 }$ and $\lambda _ { 2 }$ [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 $\operatorname {Fun}_q ( G ( k , n ) ) \rightarrow \mathbf C $ [a5].

References

[a1] E. Taft, J. Towber, "Quantum deformations of flag schemes and Grassmann schemes I. A $q$-deformation for the shape algebra $\operatorname{GL} ( n )$" 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, $q$-Series and Related Topics , Fields Inst. Commun. , 14 , Amer. Math. Soc. (1997) pp. 167–177
How to Cite This Entry:
Quantum Grassmannian. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Quantum_Grassmannian&oldid=50165
This article was adapted from an original article by Pavel Štovíček (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article