Graded quantum cluster algebras and an application to quantum Grassmannians (Q2922880)

A cluster algebra, as invented by \textit{S. Fomin} and \textit{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 [\textit{A. Berenstein} and \textit{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.NEWLINENEWLINEIt 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 \textit{J. S. Scott} [Proc. Lond. Math. Soc. (3) 92, No. 2, 345--380 (2006; Zbl 1088.22009)].NEWLINENEWLINEIn 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.