PBW basis and canonical basis

Question

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.

Answer 1

It is possible to obtain the canonical basis from the PBW basis, as long as you are working in the quantum group U_q(n). The canonical basis seems to be inherently a quantum phenomenon, so it shouldn't be surprising that we want to compute in the quantum group.

If E_π is an indexing of the quantum PBW basis, then what one first proves is that applying the bar involution is unitriangular with respect to this basis. There is even a mathoverflow question about this: Convex PBW bases.

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