I've been looking at an example in the non-commutative geometry literature and I'm having trouble figuring out what the classical motivation is. I'll just describe the classical case here: Recall that $\mathbb{CP}^2 = SU(\mathbb{C},3)/SU(\mathbb{C},2)$. Recall also that since $T_\mathbb{C}^*(\mathbb{CP}^2)$ can be viewed as a vector bundle associated to the $SU(2)$-bundle $SU(\mathbb{C},3) \to \mathbb{CP}^2$, we can view $\Omega_\mathbb{C}^1(\mathbb{CP}^2)$ as a subset  of  $\mathcal{O}(\mathbb{CP}^2) \times \mathcal{O}(\mathbb{CP}^2)$. Now in the example, the action of each of the Dolbeault operators $\partial,\overline{\partial}$ is given in terms of two mappings, $\partial_1,\partial_2$ and $\overline{\partial_1},\overline{\partial_2}$, such that 
$$
\partial(f) = (\partial_1(f),\partial_2(f)) \in \mathcal{O}(\mathbb{CP}^2) \times \mathcal{O}(\mathbb{CP}^2),
$$
and similarly for $\overline{\partial}$. The mappings ${\partial}_i, \overline{\partial}_i$ are constructed using an action of the Lie algebra $\mathfrak{su}(3)$ on $\mathcal{O}(SU(3))$ constructed using the canonical pairing between Lie algebras and coordinate algebras.
%$$
%\overline{\partial}(u^i_j) = \sum_{k=1}^3 u^i_k(
%$$
I have a feeling this is a complicated incarnation of a simple classical object. Does any of this ring any bells with anyone?

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