A cluster algebra, as invented by S. Fomin and A. Zelevinsky [J. Am. Math. Soc. 15, No. 2, 497–529 (2002; Zbl 1021.16017)], is a commutative algebra generated by a family of generators called cluster variables. The quantum deformations were defined in [A. Berenstein and A. Zelevinsky, Adv. Math. 195, No. 2, 405–455 (2005; Zbl 1124.20028)] to serve as an algebraic framework for the study of dual canonical bases in coordinate rings and their \(q\)-deformations.
It has been recognised that cluster algebra structures on homogeneous coordinate rings on Grassmannians are among the most important classes of examples. The demonstration of the existence of such a structure is due to J. S. Scott [Proc. Lond. Math. Soc. (3) 92, No. 2, 345–380 (2006; Zbl 1088.22009)].
In this paper, the authors introduce a framework for \(\mathbb{Z}\)-gradings on cluster algebras (and their quantum analogues) compatible with mutation by choosing the degrees of the (quantum) cluster variables in an initial seed subject to a compatibility with the initial exchange matrix, and then extending this to all cluster variables by mutation. In the quantum setting, by using this grading framework, the authors give a construction that produces a new quantum cluster algebra with the same cluster combinatorics but with different quasi-commutation relations between the cluster variables. As an application, the authors show that the quantum Grassmannians \(\mathbb{K}_q[\mathrm{Gr}(k,n)]\) admit quantum cluster algebra structures.