Greedy bases in rank 2 quantum cluster algebras
  We identify a quantum lift of the greedy basis for rank 2 coefficient-free cluster algebras. Our main result is that our construction does not depend on the choice of initial cluster, that it builds all cluster monomials, and that it produces bar-invariant elements. We also present several conjectures related to this quantum greedy basis and the triangular basis of Berenstein and Zelevinsky. 
Kyungyong Lee, Li Li, Dylan Rupel, and Andrei Zelevinsky
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