In noncommutative algebraic geometry a commonly studied family of objects are quantum projective spaces. Theses are certain deformations of the homogeneous coordinate ring of $\mathbb{CP}^n$. For example, see this mathoverflow post. The obvious question I would like to ask is whether or not people consider a Grassmannian generalisation of such objects, and if so, what are some well-known references.
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1$\begingroup$ For a notion of a noncommutative Grassmannian, see the MO question mathoverflow.net/questions/168993/… $\endgroup$– Pedro Lauridsen RibeiroCommented Mar 17, 2016 at 22:53
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$\begingroup$ These discussions are all valid approaches to quantum grassmannians. But more similar to the quantum projective space is the notion of quantum flag varieties--I.e. Those coming from quantum groups--which I worked out the gluing for the affine patches in my thesis... $\endgroup$– B. BischofCommented Mar 18, 2016 at 0:20
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2 Answers
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apparently, quantum grassmannians come in many variations --- this may be what you are looking for:
Graded quantum cluster algebras and an application to quantum Grassmannians, Grabowksi & Launois, 2010.
Among those who study quantized coordinate rings, it is widely acknowledged that Grassmannians have a special place. The intricate geometric structures associated to Grassmannians, due in part to their Lie-theoretic origins, give a rich structure of their quantized coordinate rings, the quantum Grassmannians.
answered Mar 17, 2016 at 19:55
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$\begingroup$ Thanks for the links Carlo! However, the second is not a noncommutative space, but something coming from quantum cohomology/string theory. The others are quantized coordinate algebras, not quantized homogeneous coordinate rings, as in the linked question. $\endgroup$ Commented Mar 17, 2016 at 22:11
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$\begingroup$ There are also various papers by Stokman and Letzter $\endgroup$– AHusainCommented Apr 16, 2016 at 23:06
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There is a deformation quantization approach to the quantization of the Grassmannians taking their Kähler symplectic form as the starting point. You can find this in the preprint by Schirmer arXiv:q-alg/9709021 from the nineties. He wrote a PhD on that. There are many earlier works on the quantization of $\mathbb{CP}^n$ in the same spirit, starting perhaps with works of Moreno and collaborators in the early eighties. You find many references in Schirmer's preprint.
However, this is not directly a $C^*$-algebraic approach, if you are interested in things like that. Nevertheless, the (a priori formal) star product is algebraic on many nice functions (the representative functions).
answered May 17, 2016 at 7:32