# Lie Algebra of an Automorphism Group Scheme over $Y$

Let $$f: X \to Y$$ be a morphism of schemes and let $$G=\operatorname{Aut}(X)$$ denote the functor sending a $$Y$$-scheme $$Z$$ to the group $$\operatorname{Aut}_Z(X \times_Y Z)$$ of automorphisms of $$X \times_Y Z$$ over $$Z$$. Let us assume $$X$$ and $$Y$$ are such that $$G$$ is a Lie group over $$Y$$. Let us denote the structure morphism by $$g: G \to Y$$.

Let $$e \in G$$ be the identity element in $$G$$ so that the Lie algebra $$\operatorname{Lie}(G) = (\mathcal{T}_{G/Y})_e$$. I saw on the Stacks page that there exists a natural identification $$(\mathcal{T}_{G/Y})_e = (\mathcal{T}_{G_s/s})_e$$ where $$s=g(e) \in Y$$, and $$G_s = G \times_Y k(s)$$ is the fiber over $$s \in Y$$.

The idea of this makes sense since tangent vector should be defined fiberwise .

However, what does $$(\mathcal{T}_{G_s/s})_e$$ represent in terms of automorphisms of $$Z$$? Is $$(\mathcal{T}_{G_s/s})_e$$ the Lie Algebra of the Lie group $$\operatorname{Aut}(X \times_Y k(s))$$ as a scheme over $$k(s) \cong k$$?