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The existence of greedy bases in rank 2 quantum cluster algebras. (English) Zbl 1397.16022
From the introduction: At the heart of the definition of quantum cluster algebras is a desire to understand nice bases in quantum algebras arising from the representation theory of nonassociative algebras. Of particular interest is the dual canonical basis in the quantized coordinate ring of a unipotent group, or more generally in the quantized coordinate ring of a double Bruhat cell. Through a meticulous study of these algebras and their bases the notion of a cluster algebra was discovered by S. Fomin and A. Zelevinsky [J. Am. Math. Soc. 15, No. 2, 497–529 (2002; Zbl 1021.16017)] with the notion of a quantum cluster algebra following in the work [Adv. Math. 195, No. 2, 405–455 (2005; Zbl 1124.20028)] of A. Berenstein and A. Zelevinsky. Underlying the definition of quantum cluster algebras is a deep conjecture that the quantized coordinate rings described above in fact have the structure of a quantum cluster algebra and that the cluster monomials arising from these cluster structures belong to the dual canonical bases of the quantum algebras. The most pressing questions in the theory are thus related to understanding bases of a (quantum) cluster algebra.
Several bases are already known for both classical (a.k.a. commutative) and quantum cluster algebras. Some of these bases do not cover the rank 2 case and some do not have the same positivity properties as the greedy basis. An incomplete list of known bases and some of their important properties is given. …
Pushing beyond affine types it was shown by K. Lee, L. Li and A. Zelevinsky in [J. Algebr. Comb. 40, No. 3, 823–840 (2014; Zbl 1316.13034)] that for wild types the set of all indecomposable positive elements can be linearly dependent and therefore does not form a basis. In contrast, these authors in [Sel. Math., New Ser. 20, No. 1, 57–82 (2014; Zbl 1295.13031)] constructed for rank 2 coefficient-free cluster algebras a combinatorially defined “greedy basis” which consists of a certain subset of the indecomposable positive elements.
Our main goal in this note is to establish the existence of a quantum lift of the greedy basis.
MSC:
16S35 Twisted and skew group rings, crossed products
13F60 Cluster algebras
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