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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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    $\begingroup$ For a notion of a noncommutative Grassmannian, see the MO question mathoverflow.net/questions/168993/… $\endgroup$ – Pedro Lauridsen Ribeiro Mar 17 '16 at 22:53
  • $\begingroup$ 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. Bischof Mar 18 '16 at 0:20
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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.

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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. $\endgroup$ – Lars Pettersen Mar 17 '16 at 22:11
  • $\begingroup$ There are also various papers by Stokman and Letzter $\endgroup$ – AHusain Apr 16 '16 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).

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