## Graded quantum cluster algebras and an application to quantum Grassmannians.(English)Zbl 1315.13036

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.

### MSC:

 13F60 Cluster algebras 20G42 Quantum groups (quantized function algebras) and their representations 17B37 Quantum groups (quantized enveloping algebras) and related deformations

### Keywords:

cluster algebras; quantum cluster algebras; Grassmannians

### Citations:

Zbl 1021.16017; Zbl 1124.20028; Zbl 1088.22009
Full Text:

### References:

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