# PBW basis and canonical basis

Consider the example of $\mathfrak{g} = sl_3$. Then $$\mathfrak{g} = \mathfrak{n} \oplus \mathfrak{h} \oplus \mathfrak{n}^{-},$$ where $\mathfrak{n}$ is generated by $E_{12}, E_{13}, E_{23}$, $\mathfrak{h}$ is generated by $E_{11}-E_{22}, E_{22}-E_{33}$, $\mathfrak{n}^{-}$ is generated by $E_{21}, E_{32}, E_{31}$, $E_{ij}$ is a matrix with $1$ at $(i,j)$ and $0$ elsewhere.

A PBW basis of $U(\mathfrak{n})$ is $$B_1 = \{ E_1^a (E_1 E_2 - E_2 E_1)^b E_2^c \mid a, b, c \in \mathbb{N} \}.$$

There is another basis of $U(\mathfrak{n})$ called canonical basis which is given by $$B_2 = \{ E_1^aE_2^bE_1^c \mid a+c \geq b \} \cup \{ E_2^aE_1^bE_2^c \mid a+c \geq b \}.$$

The basis $B_2$ has a property: given a lowest weight vector $v_0$ of a representation $V$ of $U(\mathfrak{n})$, the set $$\{ b v_0 \mid b v_0 \ne 0 \}$$ is a basis of $V$.

My question is: in general, how to compute canonical basis for $U(\mathfrak{n})$ explicitly like the above example. Can we derive a canonical basis from a PBW basis? Thank you very much.

• Can you please say what is $E_1$ and $E_2$ in the given bases? The original bases for the Lie algebra is in terms of $E_{i,j}$, so these bases also should be in terms of $E_{i,j}$ no? why in terms of $E_1$ and $E_2$ ? – GA316 Jan 24 '17 at 5:57
• @GA316, sorry, I used different notations. Here $E_1=E_{12}$, $E_{2}=E_{23}$. – Jianrong Li Jan 24 '17 at 9:05

Now it is a matter of linear algebra to prove that there exists a unique bar invariant basis bπ such that $$b_\pi=\sum_\sigma c_{\pi\sigma} E_\sigma$$ with cσσ=1, cσπ∈qℤ[q] if σ≠π and cσπ=0 unless σ≤π. This basis bπ is the canonical basis.