My question is inspired by quantum spheres and quantum projective spaces (studied by Dijkhuizen & Noumi, Vaksman & Soibelman, etc., i.e. "A FAMILY OF QUANTUM PROJECTIVE SPACES AND RELATED $q$-HYPERGEOMETRIC ORTHOGONAL POLYNOMIALS" and "The algebra of functions on the quantum group $SU(n+1)$ and odd-dimensional quantum spheres"). My question is: how to obtain the algebra they introduced there. It may be an ad-hoc construction, but I am sure that there exists (not a functiorial) construction of such quantum algebras. For example I want to gerenalize the concept of the algebra of functions on the total space of a family of quantum $(2n−1)$-spheres to quantum Grassmannian. Therefore to be more explicit: Given a (complex) Grassmannian $Gr(n,m)$, how to define the algebra of functions on the total space of a family of quantum Grassmannians? One can use the definition o f S.Launois and T.H.Lenagan given in "Twisting the quantum grassmannian" to define such algebras, but from that we will obtain (I suspect so) the, what we call, standard quantum Grassmannians (in particular for the cases $Gr(n,n+1)$ the standard quantum projective spaces without any parameter $c$ and $d$). I hope you understand what my goal is? Does someone have an idea about that? Thanks a lot.
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1$\begingroup$ I don't think I do understand what your goal is. Is it that you want quantizations of Grassmannians that are different from the standard one? What would qualify as a satisfactory answer? (I hesitate slightly to throw in the oft repeated but still true comment that in general it's hard, in the mathematical sense, to just "quantize an algebra of functions". There are lots of different approaches in the literature - saying which you've read and which did or didn't qualify in your context would help.) $\endgroup$– Jan GrabowskiCommented Mar 3, 2016 at 16:14
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$\begingroup$ I don't understand what your goal is either, but look at the papers of Gail Letzter too. $\endgroup$– AHusainCommented Mar 3, 2016 at 19:18
1 Answer
The family of quantum projective spaces you mention can be constructed in two different (equivalent) ways starting from the quantum group $SU_q(n)$.
In itself this family is a family of $*$-subalgebras and right coideals inside $SU_q(n)$; this is what is called a quantum homogeneous space.
You can either find it as invariant elements with respect to a suitable right ideal and two-sided coideal inside ${\cal U}_q(\mathfrak{su}(n)$ or as coinvariants w.r. to a coalgebra projection $\pi:SU_q(n)\to C$, $C$ beign a pointed coalgebra (this is less developed in the literature and it is what is called a quantum coisotropic subgroup). That the two approaches are equivalent is basically an instance of what is called quantum duality principle for quantum homogeneous spaces.
I am thinking at algebraic approaches and such approaches will provide you an algebra with generators and relations. However in this cases everything fits well as to produce also a $C^*$--algebra quantization. The $C^*$-algebra quantization can be obtained also via different quantization techniques (look for "non standard quantum complex projective spaces").
The algebraic part certainly generalizes to a family of complex grassmannians, though I never saw writing generators and relations explicitely (guess it is long and painful...). But since Letzter's work classifies right coideal subalgebras in compact quantum groups it should be hidden somewhere in her papers.
