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

## TL;DR

This paper proves the cluster multiplication theorem for the quantum cluster algebra of type A_{2}^{(2)}, leading to bar-invariant bases and confirming positivity of elements within these bases.

## Contribution

It establishes the cluster multiplication theorem for this specific quantum cluster algebra and explicitly describes a bar-invariant basis, advancing understanding of its structure.

## Key findings

- Bar-invariant Z[q^{±1/2}]-bases are constructed
- Positivity of elements in these bases is confirmed
- A triangular basis for the algebra is explicitly described

## 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.