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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.

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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.

For finite type the canonical basis is all cluster monomials and for affine type the basis is all cluster monomials along with some additional elements.

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