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A [[unital algebra]] $A$ that is a [[co-module]] for a quantum group $\operatorname{Fun}_{q} ( G )$ (cf. [[Quantum groups|Quantum groups]]) and for which the structure mapping $L : A \rightarrow \operatorname { Fun } _ { q } ( G ) \otimes A$ is an algebra homomorphism, i.e., $A$ is a co-module algebra [[#References|[a1]]]. Here, $\operatorname{Fun}_{q} ( G )$ is a deformation of the [[Poisson algebra|Poisson algebra]] $\operatorname{Fun}( G )$, of a Poisson–Lie group $G$, endowed with the structure of a Hopf algebra with a co-multiplication $\Delta$ and a co-unit <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q120/q120030/q1200309.png"/>. Often, both $A$ and $\operatorname{Fun}_{q} ( G )$ can also be equipped with a $*$-involution. The left co-action $L$ satisfies
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A [[unital algebra]] $A$ that is a [[co-module]] for a quantum group $\operatorname{Fun}_{q}(G)$ (cf. [[Quantum groups]]) and for which the structure mapping $L : A \rightarrow \operatorname { Fun }_{q} (G) \otimes A$ is an algebra homomorphism, i.e., $A$ is a co-module algebra [[#References|[a1]]]. Here, $\operatorname{Fun}_{q} (G)$ is a deformation of the [[Poisson algebra]] $\operatorname{Fun}(G)$, of a Poisson–Lie group $G$, endowed with the structure of a Hopf algebra with a co-multiplication $\Delta$ and a co-unit $\epsilon$. Often, both $A$ and $\operatorname{Fun}_{q} (G)$ can also be equipped with a $*$-involution. The left co-action $L$ satisfies
  
<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/q120030/q12003015.png"/></td> </tr></table>
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\begin{equation*} ( \operatorname{id}_{\operatorname{Fun}(G)} \bigotimes L ) \circ L = ( \Delta \bigotimes \operatorname{id} _ { A } ) \circ L. \end{equation*}
  
 
\begin{equation*} ( \epsilon \bigotimes \operatorname{id} _ { A } ) \circ L = \operatorname{id} _ { A }. \end{equation*}
 
\begin{equation*} ( \epsilon \bigotimes \operatorname{id} _ { A } ) \circ L = \operatorname{id} _ { A }. \end{equation*}
  
These relations should be modified correspondingly for a right co-action. In the dual picture, if ${\cal U} _ { q } ( \mathfrak { g } )$ is the deformed universal enveloping algebra of the Lie algebra $\frak g$ and $\langle \, .\, ,\,  . \, \rangle$ is a non-degenerate dual pairing between the Hopf algebras ${\cal U} _ { q } ( \mathfrak { g } )$ and $\operatorname{Fun}_{q} ( G )$, then the prescription $X\cdot f = ( \langle X , \cdot \rangle \otimes \operatorname {id} _ { A } ) L ( f )$, with $X \in \mathcal U _ { q } ( \mathfrak { g } )$ and $f \in A$, defines a right action of ${\cal U} _ { q } ( \mathfrak { g } )$ on $A$ ($X.( Y . f ) = ( Y X ) . f$) and one has
+
These relations should be modified correspondingly for a right co-action. In the dual picture, if ${\cal U}_{q} (\mathfrak { g })$ is the deformed universal enveloping algebra of the Lie algebra $\frak g$ and $\langle \, .\, ,\,  . \, \rangle$ is a non-degenerate dual pairing between the Hopf algebras ${\cal U} _ { q } ( \mathfrak { g } )$ and $\operatorname{Fun}_{q} ( G )$, then the prescription $X\cdot f = ( \langle X , \cdot \rangle \otimes \operatorname {id} _ { A } ) L ( f )$, with $X \in \mathcal U _ { q } ( \mathfrak { g } )$ and $f \in A$, defines a right action of ${\cal U} _ { q } ( \mathfrak { g } )$ on $A$ ($X.( Y . f ) = ( Y X ) . f$) and one has
  
 
\begin{equation*} X. ( f g ) = \mu ( \Delta X . ( f \bigotimes g ) ), \end{equation*}
 
\begin{equation*} X. ( f g ) = \mu ( \Delta X . ( f \bigotimes g ) ), \end{equation*}
  
where $\mu : A \otimes A \rightarrow A$ is the multiplication in $A$ and $\Delta$ is the co-multiplication in $\mathcal{U} ( \mathfrak { g } )$. Typically, $A$ is a deformation of the Poisson algebra $\operatorname { Fun } ( M )$ (frequently called the quantization of $M$), where $M$ is a Poisson manifold and, at the same time, a left homogeneous space of $G$ with the left action $G \times M \rightarrow M$ a Poisson mapping.
+
where $\mu : A \otimes A \rightarrow A$ is the multiplication in $A$ and $\Delta$ is the co-multiplication in $\mathcal{U} ( \mathfrak { g } )$. Typically, $A$ is a deformation of the Poisson algebra $\operatorname{Fun}(M)$ (frequently called the quantization of $M$), where $M$ is a Poisson manifold and, at the same time, a left homogeneous space of $G$ with the left action $G \times M \rightarrow M$ a Poisson mapping.
  
It is not quite clear how to translate into purely algebraic terms the property that $M$ is a homogeneous space of $G$. One possibility is to require that only multiples of the unit $1 \in A$ satisfy $L ( f ) = 1 \otimes f$. A stronger condition requires the existence of a linear functional $\varphi \in A ^ { * }$ such that $\varphi ( 1 ) = 1$ while the linear mapping $\psi = ( \text { id } \otimes \varphi ) \circ L : A \rightarrow \operatorname { Fun } _ { q } ( G )$ be injective. Then $\varphi$ can be considered as a base point.
+
It is not quite clear how to translate into purely algebraic terms the property that $M$ is a homogeneous space of $G$. One possibility is to require that only multiples of the unit $1 \in A$ satisfy $L(f) = 1 \otimes f$. A stronger condition requires the existence of a linear functional $\varphi \in A ^ { * }$ such that $\varphi(1) = 1$ while the linear mapping $\psi = ( \text { id } \otimes \varphi ) \circ L : A \rightarrow \operatorname { Fun } _ { q } ( G )$ be injective. Then $\varphi$ can be considered as a base point.
  
 
The still stronger requirement that, in addition, $\varphi$ be a homomorphism (a so-called classical point) holds when $A$ is a quantization of a Poisson homogeneous space $M = G / H$ with $H \subset G$ a Poisson–Lie subgroup. The quantum homogeneous space $\operatorname { Fun } _ { q } ( G / H )$ is defined as the subalgebra in $\operatorname{Fun}_{q} ( G )$ formed by $H$-invariant elements $f$, $( \text { id } \otimes \pi ) \Delta f = f \otimes 1$ where $\pi : \operatorname { Fun } _ { q } ( G ) \rightarrow \operatorname { Fun } _ { q } ( H )$ is a Hopf-algebra homomorphism.
 
The still stronger requirement that, in addition, $\varphi$ be a homomorphism (a so-called classical point) holds when $A$ is a quantization of a Poisson homogeneous space $M = G / H$ with $H \subset G$ a Poisson–Lie subgroup. The quantum homogeneous space $\operatorname { Fun } _ { q } ( G / H )$ is defined as the subalgebra in $\operatorname{Fun}_{q} ( G )$ formed by $H$-invariant elements $f$, $( \text { id } \otimes \pi ) \Delta f = f \otimes 1$ where $\pi : \operatorname { Fun } _ { q } ( G ) \rightarrow \operatorname { Fun } _ { q } ( H )$ is a Hopf-algebra homomorphism.
  
A richer class of examples is provided by quantization of orbits of the dressing transformation of $G$, acting on its dual Poisson–Lie group (also called the generalized Pontryagin dual) $G ^ { * }$. The best studied cases concern the compact and solvable factors $K$ and $AN$ ($K$ and $AN$ are mutually dual) in the Iwasawa decomposition $\mathfrak { G } = K.AN$, where $\mathfrak{G}$ is a simple complex [[Lie group|Lie group]]. One obtains this way, among others, the [[Quantum sphere|quantum sphere]] and, more generally, [[Quantum Grassmannian|quantum Grassmannian]] and quantum flag manifolds.
+
A richer class of examples is provided by quantization of orbits of the dressing transformation of $G$, acting on its dual Poisson–Lie group (also called the generalized Pontryagin dual) $G^{*}$. The best studied cases concern the compact and solvable factors $K$ and $AN$ ($K$ and $AN$ are mutually dual) in the Iwasawa decomposition $\mathfrak { G } = K.AN$, where $\mathfrak{G}$ is a simple complex [[Lie group|Lie group]]. One obtains this way, among others, the [[Quantum sphere|quantum sphere]] and, more generally, [[quantum Grassmannian]] and quantum flag manifolds.
  
 
There is a vast amount of literature on this subject. The survey book [[#References|[a2]]] contains a rich list of references.
 
There is a vast amount of literature on this subject. The survey book [[#References|[a2]]] contains a rich list of references.
  
 
====References====
 
====References====
<table><tr><td valign="top">[a1]</td> <td valign="top">  E. Abe,   "Hopf algebras" , Cambridge Univ. Press  (1977)</td></tr><tr><td valign="top">[a2]</td> <td valign="top">  V. Chari,   A. Pressley,   "A guide to quantum groups" , Cambridge Univ. Press  (1994)</td></tr></table>
+
<table>
 +
<tr><td valign="top">[a1]</td> <td valign="top">  E. Abe, "Hopf algebras" , Cambridge Univ. Press  (1977)</td></tr>
 +
<tr><td valign="top">[a2]</td> <td valign="top">  V. Chari, A. Pressley, "A guide to quantum groups" , Cambridge Univ. Press  (1994)</td></tr>
 +
</table>

Latest revision as of 13:24, 25 November 2023

A unital algebra $A$ that is a co-module for a quantum group $\operatorname{Fun}_{q}(G)$ (cf. Quantum groups) and for which the structure mapping $L : A \rightarrow \operatorname { Fun }_{q} (G) \otimes A$ is an algebra homomorphism, i.e., $A$ is a co-module algebra [a1]. Here, $\operatorname{Fun}_{q} (G)$ is a deformation of the Poisson algebra $\operatorname{Fun}(G)$, of a Poisson–Lie group $G$, endowed with the structure of a Hopf algebra with a co-multiplication $\Delta$ and a co-unit $\epsilon$. Often, both $A$ and $\operatorname{Fun}_{q} (G)$ can also be equipped with a $*$-involution. The left co-action $L$ satisfies

\begin{equation*} ( \operatorname{id}_{\operatorname{Fun}(G)} \bigotimes L ) \circ L = ( \Delta \bigotimes \operatorname{id} _ { A } ) \circ L. \end{equation*}

\begin{equation*} ( \epsilon \bigotimes \operatorname{id} _ { A } ) \circ L = \operatorname{id} _ { A }. \end{equation*}

These relations should be modified correspondingly for a right co-action. In the dual picture, if ${\cal U}_{q} (\mathfrak { g })$ is the deformed universal enveloping algebra of the Lie algebra $\frak g$ and $\langle \, .\, ,\, . \, \rangle$ is a non-degenerate dual pairing between the Hopf algebras ${\cal U} _ { q } ( \mathfrak { g } )$ and $\operatorname{Fun}_{q} ( G )$, then the prescription $X\cdot f = ( \langle X , \cdot \rangle \otimes \operatorname {id} _ { A } ) L ( f )$, with $X \in \mathcal U _ { q } ( \mathfrak { g } )$ and $f \in A$, defines a right action of ${\cal U} _ { q } ( \mathfrak { g } )$ on $A$ ($X.( Y . f ) = ( Y X ) . f$) and one has

\begin{equation*} X. ( f g ) = \mu ( \Delta X . ( f \bigotimes g ) ), \end{equation*}

where $\mu : A \otimes A \rightarrow A$ is the multiplication in $A$ and $\Delta$ is the co-multiplication in $\mathcal{U} ( \mathfrak { g } )$. Typically, $A$ is a deformation of the Poisson algebra $\operatorname{Fun}(M)$ (frequently called the quantization of $M$), where $M$ is a Poisson manifold and, at the same time, a left homogeneous space of $G$ with the left action $G \times M \rightarrow M$ a Poisson mapping.

It is not quite clear how to translate into purely algebraic terms the property that $M$ is a homogeneous space of $G$. One possibility is to require that only multiples of the unit $1 \in A$ satisfy $L(f) = 1 \otimes f$. A stronger condition requires the existence of a linear functional $\varphi \in A ^ { * }$ such that $\varphi(1) = 1$ while the linear mapping $\psi = ( \text { id } \otimes \varphi ) \circ L : A \rightarrow \operatorname { Fun } _ { q } ( G )$ be injective. Then $\varphi$ can be considered as a base point.

The still stronger requirement that, in addition, $\varphi$ be a homomorphism (a so-called classical point) holds when $A$ is a quantization of a Poisson homogeneous space $M = G / H$ with $H \subset G$ a Poisson–Lie subgroup. The quantum homogeneous space $\operatorname { Fun } _ { q } ( G / H )$ is defined as the subalgebra in $\operatorname{Fun}_{q} ( G )$ formed by $H$-invariant elements $f$, $( \text { id } \otimes \pi ) \Delta f = f \otimes 1$ where $\pi : \operatorname { Fun } _ { q } ( G ) \rightarrow \operatorname { Fun } _ { q } ( H )$ is a Hopf-algebra homomorphism.

A richer class of examples is provided by quantization of orbits of the dressing transformation of $G$, acting on its dual Poisson–Lie group (also called the generalized Pontryagin dual) $G^{*}$. The best studied cases concern the compact and solvable factors $K$ and $AN$ ($K$ and $AN$ are mutually dual) in the Iwasawa decomposition $\mathfrak { G } = K.AN$, where $\mathfrak{G}$ is a simple complex Lie group. One obtains this way, among others, the quantum sphere and, more generally, quantum Grassmannian and quantum flag manifolds.

There is a vast amount of literature on this subject. The survey book [a2] contains a rich list of references.

References

[a1] E. Abe, "Hopf algebras" , Cambridge Univ. Press (1977)
[a2] V. Chari, A. Pressley, "A guide to quantum groups" , Cambridge Univ. Press (1994)
How to Cite This Entry:
Quantum homogeneous space. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Quantum_homogeneous_space&oldid=49876
This article was adapted from an original article by Pavel Štovíček (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article