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It was realized recently that on-shell diagrams are objects of fundamental importance in the analysis of scattering amplitudes. In four-dimensional theories planar on-shell diagrams are closely related to the positive Grassmannian and the cell decomposition of it into the union of so called positroid cells. (The most complete results were obtained for planar diagrams of maximally supersymmetric Yang-Mills theory, but the relation to positive Grassmannian exists also in other cases.) From the other side it was discovered that there exists a bijective correspondence between positroid cells in positive Grassmannian and prime ideals in quantum Grassmannian that are invariant with respect to the torus action. We come to the idea that the theory of planar off-shell diagrams can be quantized (more precisely, q-deformed); one can hope that the quantization can lead to physically interesting results.

We prove some results in this direction. Namely, we establish that volume forms on positroids used in the representation of amplitudes can be q-deformed to Hochschild homology classes of corresponding quantum algebras