Quantum homogeneous space

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

$$({\mathrm{id}}_{\mathrm{Fun}(G)}⨂L)\circ L=(\mathrm{\Delta }⨂{\mathrm{id}}_A)\circ L.$$

$$(ϵ⨂{\mathrm{id}}_A)\circ L={\mathrm{id}}_A.$$

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

$$X.(fg)=\mu (\mathrm{\Delta }X.(f⨂g)),$$

where $\mu :A\otimes A\to A$ is the multiplication in $A$ and $\mathrm{\Delta }$ is the co-multiplication in $\mathcal{U}(\mathfrak{g})$. Typically, $A$ is a deformation of the Poisson algebra $\mathrm{Fun}(M)$ (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 $G\times M\to M$ 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 $\phi \in A^{\ast }$ such that $\phi (1)=1$ while the linear mapping $\psi =(\text{ id }\otimes \phi )\circ L:A\to {\mathrm{Fun}}_q(G)$ be injective. Then $\phi$ can be considered as a base point.

The still stronger requirement that, in addition, $\phi$ 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 ${\mathrm{Fun}}_q(G/H)$ is defined as the subalgebra in ${\mathrm{Fun}}_q(G)$ formed by $H$-invariant elements $f$, $(\text{ id }\otimes \pi )\mathrm{\Delta }f=f\otimes 1$ where $\pi :{\mathrm{Fun}}_q(G)\to {\mathrm{Fun}}_q(H)$ 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) $G^{\ast }$. 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.

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