# Definition of action of Lie algebra of an algbraic group

Here is the context of my interrogation :

Let $$G$$ be an affine algebraic group over $$\mathbb{C}$$ acting rationally on an affine variety $$X$$ over $$\mathbb{C}$$.

This induces an action of $$G$$ on $$\mathbb{C}[X]$$ the algebra of regular functions on $$X$$ by: $$\begin{array}{cccl} \lambda :& G \times \mathbb{C}[X] & \to & \mathbb{C}[X] \newline & (g,f) & \mapsto & g \cdot f : x \mapsto f(g^{-1}x) \end{array}$$ This is an action by $$\mathbb{C}$$-algebra homomorphism (because for all $$g \in G$$, $$\lambda_g : f \mapsto \lambda(g,f)$$ is a $$\mathbb{C}$$-algebra homomorphism).

Now, let $$Lie(G)$$ the Lie algebra of $$G$$ that is the set of derivations $$\delta$$ on $$\mathbb{C}[G]$$ such that $$\forall g \in G, \lambda_g \circ \delta = \delta \circ \lambda_g.$$ (We denoted $$\lambda_g : \mathbb{C}[G] \to \mathbb{C}[G]$$ with the same symbol as the one above : $$\lambda_g : \mathbb{C}[X] \to \mathbb{C}[X]$$ in order not to write twice the same idea).

I wanted ton define an action of $$Lie(G)$$ on $$\mathbb{C}[X]$$ by derivation, but I don't know what is the definition of a Lie algebra action in the context of algebraic geometry. I searched everywhere for a definition, but only found the one in the context of differential geometry, which can't apply here.

Thanks in advance for any hint, help or reference for the defintion.

K. Y.