# Canonical basis of cluster algebras

Let $x_{k+1} = \frac{x_k^{d_k}+1}{x_{k-1}}$, $k \in \mathbb{Z}$, where $d_{k+2} = d_k \in \mathbb{Z_{>0}}$. Let $b=d_1$ and $c=d_2$. Define the cluster algebra $A = A(\left( \begin{matrix} 0 & b \\ -c & 0 \end{matrix} \right))$ to be the algebra generated by all $x_k$. Is the following set $B$ a totally positive basis of $A$? $$B = \{ x_0^{m_0} x_1^{m_1} x_2^{m_2} x_3^{m_3}: min(m_0, m_2)=0, min(m_1, m_3)=0 \}.$$ Are there some references for the canonical basis of the algebra $A$? Thank you very much.

You should look at Positivity and canonical bases in rank 2 cluster algebras of finite and affine types by Sherman and Zelevinsky where they are able to construct a canonical basis explicitly for finte type and affine type (i.e. $bc < 4$ and $bc = 4$ respectively). Your $B$ is a basis for $A$ called the standard monomial basis. However it is not canonical as you can choose any four consecutive $x_i, x_{i+1}, x_{i+2}, x_{i + 3}$. Also it does not have the following notion of positivity.
Given $y \in A$ we call $y$ positive if the expansion of $y$ as a Laurent polynomial in every cluster $x_i, x_{i+1}$ has only positive coefficients. We want a basis so that the positive elements of $A$ are positive combinations of the basis elements.
For $(b,c) = (2,2)$ consider $z = x_0x_3 - x_1x_2$, then $$z = \frac{x_1^2 + x_2^2 + 1}{x_1 x_2},$$ and by symmetry is also a Laurent polynomial with positive coefficients for any other cluster.