Quantum homogeneous space

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)