Quantum homogeneous space

A unital algebra that is a co-module for a quantum group (cf. Quantum groups) and for which the structure mapping is an algebra homomorphism, i.e., is a co-module algebra [a1]. Here, is a deformation of the Poisson algebra , of a Poisson–Lie group , endowed with the structure of a Hopf algebra with a co-multiplication and a co-unit . Often, both and can also be equipped with a -involution. The left co-action satisfies

These relations should be modified correspondingly for a right co-action. In the dual picture, if is the deformed universal enveloping algebra of the Lie algebra and is a non-degenerate dual pairing between the Hopf algebras and , then the prescription , with and , defines a right action of on () and one has

where is the multiplication in and is the co-multiplication in . Typically, is a deformation of the Poisson algebra (frequently called the quantization of ), where is a Poisson manifold and, at the same time, a left homogeneous space of with the left action a Poisson mapping.

It is not quite clear how to translate into purely algebraic terms the property that is a homogeneous space of . One possibility is to require that only multiples of the unit satisfy . A stronger condition requires the existence of a linear functional such that while the linear mapping be injective. Then can be considered as a base point.

The still stronger requirement that, in addition, be a homomorphism (a so-called classical point) holds when is a quantization of a Poisson homogeneous space with a Poisson–Lie subgroup. The quantum homogeneous space is defined as the subalgebra in formed by -invariant elements , where is a Hopf-algebra homomorphism.

A richer class of examples is provided by quantization of orbits of the dressing transformation of , acting on its dual Poisson–Lie group (also called the generalized Pontryagin dual) . The best studied cases concern the compact and solvable factors and ( and are mutually dual) in the Iwasawa decomposition , where 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)