Bases of the quantum cluster algebra of the Kronecker quiver

Details Bases of the quantum cluster algebra of the Kronecker quiver Bases of the quantum cluster algebra of the Kronecker quiver

2010-04-14

arxiv.org/abs/1004.2349v4

Mathematics

Representation Theory

12 pages. Add a section to give an representation-theoretic interpretation of the element $s_n$.

Scientific paper

We construct bar-invariant $\mathbb{Z}[q^{\pm 1/2}]-$bases of the quantum cluster algebra of the Kronecker quiver which are quantum analogues of the canonical basis, semicanonical basis and dual semicanonical basis of the cluster algebra of the Kronecker quiver in the sense of \cite{sherzel},\cite{calzel} and \cite{gls} respectively. As a byproduct, we prove the positivity of the elements in these bases.

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