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.


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