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](https://mathoverflow.net/questions/109347/point-modules-of-quantum-projective-space-mathbbpn). 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.



Quantum Grassmannians?
apparently, [quantum grassmannians](https://www.encyclopediaofmath.org/index.php/Quantum_Grassmannian) come in many variations --- this may be what you are looking for:


[Graded quantum cluster algebras and an application to quantum Grassmannians](http://arxiv.org/abs/1301.2133), 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*.
> 
> 
> 


