Let $A=(a\_{ij})$ be a generalized Cartan matrix of order $n$ and $D=diag(d\_1,\ldots,d\_n)$ the diagonal matrix such that $DA$ is symmetric. Let $$E\_{ij}=\sum\_{r+s=1-a\_{ij}} (-1)^r E\_i^{(r)} E\_j E\_i^{(s)},$$ $i,j\in \{1, \ldots, n\}$, $E\_i^{(k)} = \frac{E\_i^k}{[k]\_{q\_i}!}$, $q\_i = q^{d\_i}$. 


Let 
$$
U\_+ = \mathbb{C}(q)\langle E\_1, \ldots, E\_n \rangle/\langle E\_{ij}: i \neq j \rangle.
$$


Let $A=\left( \begin{matrix} 2 & -1 \\ -1 & 2 \end{matrix} \right)$. How to show that the set $$B = \{E\_1^{(a)}E\_2^{(b)}E\_1^{(c)}, E\_2^{(c)}E\_1^{(b)}E\_2^{(a)}: b \geq a+c\}$$ is a canonical basis for $U\_+$? Where does the condition $b \geq a+c$ come from? Thank you very much.



Examples of canonical bases
The complete calculation is done in [Lusztig's book](https://books.google.de/books/about/Introduction_to_Quantum_Groups.html?id=HKPjCUiOUQ0C&redir_esc=y&hl=en) (Lemma 42.1.2). Essentially, the condition $b\geq a+c$ comes from the Serre relations; e.g. we have
$$E\_1E\_2E\_1=E\_1^{(2)}E\_2+E\_2E\_1^{(2)}.$$


