Let $X$ be a scheme and $x$ a closed point. Let us call $\mathcal{I_x}$ the sheaf of ideals in $\mathcal{O}_X$ that defines $x$ Notice that the canonical embedding $x \hookrightarrow X$ is given by the sheaf map $\mathcal{O}_X \to \mathcal{O}_X/\mathcal{I_x}$. Now consider the ideal $\mathcal{I}_x^2$ and the closed subscheme defined by it. It is supported in $ x $ but it has as structure sheaf an Artin algebra with generators of degree two that essentially displays the infinitesimal information of $X$ around $x$, in particular, it contains the tangent space. In the case of an abelian variety this subscheme should determine the Lie algebra.

If $X$ is finite type over a field, say, then if $X$ is smooth then all the infinitesimal information is contained in this first order infinitesimal neighborhood. If $X$ is singular at $x$ you need the full "tangent cone" and to recover it from local data you need to pass to the limit of $\operatorname{Spec}(\mathcal{O}_X/\mathcal{I}_x^n)$ over $n \geq 1$ to get a formal scheme as suggested by Mariano.

If $X$ is an abelian variety, the completion of $X$ along the neutral point gives a formal group with topological dimension 0 but with "hidden dimension" that of $X$. To get more information about this, search for *formal groups*, there are several resources online an in print, especially formal groups associated to abelian varieties.