Let $ X=\mathbb{P}_k^1$ and let $ X_{k[\epsilon]} = X \times_k \operatorname{Spec} k[\epsilon]/(\epsilon)^2$ denote the trivial first order deformation of $ X$ over $ k[\epsilon]/(\epsilon)^2$ .

Let $ G= PGL(2,k)$ so that we have the following string of isomorphisms $ $ sl_2 \cong \operatorname{Lie}(G) \cong (\mathcal{T}G)_{e} \cong G(k[\epsilon])_{(\epsilon) \to e} = \operatorname{Aut}_{k[\epsilon]}(X_{k[\epsilon]}).$ $

If $ \phi \in \operatorname{Aut}_{k[\epsilon]}(X_{k[\epsilon]})$ , then $ \phi$ is locally of the form, $ $ z \mapsto z + \epsilon(a_0 + a_1 z + a_2 z^2)$ $ where $ z$ is a choice of homogeneous coordinate on $ X$ .

I am trying to recover $ sl_2$ from this local description. If I let $ a_0=b, a_1= a-d$ , and $ a_2= -c$ , then $ $ \frac{(1+a\epsilon)z+b\epsilon}{c\epsilon z + (1+d\epsilon)}=z+\epsilon\big( b+(a-d)z-cz^2 \big), $ $ And, letting $ z=v/u$ , we can find that $ $ u \mapsto u + \epsilon(d u + cv)$ $ $ $ v \mapsto v + \epsilon(bu + av)$ $

But I don’t see how this relates to $ sl_2$ . Is it even possible for one to recover $ sl_2$ from this description?