Triangular bases in quantum cluster algebras (Q2878722)

Cluster algebras were invented by \textit{S. Fomin} and \textit{A. Zelevinsky} [J. Am. Math. Soc. 15, No. 2, 497--529 (2002; Zbl 1021.16017)] in order to study total positivity in algebraic groups and canonical bases in quantum groups. A lot of recent activity has been directed towards various constructions of ``natural'' bases in cluster algebras for a better understanding of canonical bases in quantum groups. In this paper, the authors developed a new approach to this problem which is close in spirit to Lusztig's original way of constructing a canonical basis, and the pioneering construction of the Kazhdan-Lusztig basis in a Hecke algebra. The key ingredient of their approach is a new version of Lusztig's lemma which was used to construct a ``natural'' basis called the triangular canonical basis of any quantum cluster algebra with an acyclic quantum seed. The authors also showed that this basis contains all cluster monomials associated to acyclic quantum seeds.