UC Riverside

We study connections between quantum cluster algebras and the double canonical bases of subalgebras of the Heisenberg and Drinfeld double associated to a quantized Borel subalgebra of $\mathfrak{sl}_3$. We show that the Heisenberg double has a finite type quantum cluster algebra structure for which the set of quantum cluster monomials is equal to the double canonical basis. Furthermore, we identify an affine quantum cluster algebra structure on parabolic subalgebras of the Drinfeld double and prove that all quantum cluster variables belong to the double canonical basis. Finally, we identify an infinite subset of quantum clusters for which the quantum cluster monomials are contained in the double canonical basis.