Cluster multiplication theorem in the quantum cluster algebra of type $A_2^{(2)}$

Abstract

The objective of the present paper is to prove cluster multiplication theorem in the quantum cluster algebra of type $A_2^{(2)}$. As corollaries, we obtain bar-invariant $\mathbb{Z}[q^{\pm \frac{1}{2}}]$-bases established in [6], and naturally deduce the positivity of the elements in these bases. One bar-invariant basis as the triangular basis of this quantum cluster algebra is also explicitly described.

36 pages, comments welcome