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Are there some standard ways to define the tangent space to a group $G$ at its unit element, $e$, when the group is not (pro)algebraic/(pro)Lie, not necessarily over a field, does not have the ``classical" topology? More precisely, I would like to have (for some large class of groups) the structure: $T_{(G,e)}\rightleftarrows G$, where $T_{(G,e)}$ is in some sense linear, and the arrows are some substitutions for the standard ln/exp maps.

For example, for a local ring $R$ (not necessarily complete), consider the group of automorphisms, $Aut(R)$. Its tangent space is expected to be the $R$-module of derivations, $Der(R)$. The traditional ln/exp are impossible in the general case (both because of non-convergence and because of e.g. positive characteristic). But some substitutions can be cooked up, at least when $R$ is a local ring with some minimal restrictions.

We have defined the tangent space (with some substitutions of ln/exp) in a particular case, when the group is filtered and acts on a filtered module, here. As always, I do not want to reinvent the wheel.

(A related question)

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    $\begingroup$ The automorphism group of a commutative ring $R$ has much more structure than a group: it has a functor of points, where its value on a ring $S$ is the automorphism group of $R \otimes_k S$ over $S$. This functor of points knows the Lie algebra of derivations of $R$, which you can recover by considering $S = k[\epsilon]/\epsilon^2$. So you are in an important sense still in the (pro)algebraic group case here, or at least a mild generalization of it. If you just have a bare group, you can always take its proalgebraic completion, but it sounds like you don't just have a bare group. $\endgroup$
    – Qiaochu Yuan
    Commented Nov 6, 2018 at 1:29
  • $\begingroup$ @Qiaochu Yuan Thanks! What is the pro-algebraic completion for an arbitrary/bare group? $\endgroup$
    – Dmitry Kerner
    Commented Nov 8, 2018 at 20:54
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    $\begingroup$ A relatively clean description is that the proalgebraic completion of a group $G$ over a field $k$ is the affine group scheme Tannaka-Krein reconstruction outputs if you input the category of finite-dimensional representations of $G$ over $k$. If $k = \mathbb{C}$, $G$ could also be a topological group and you could consider continuous representations. See also math.stackexchange.com/questions/1163055/… . $\endgroup$
    – Qiaochu Yuan
    Commented Nov 8, 2018 at 21:22
  • $\begingroup$ Thanks. Still this is not the full answer :( $\endgroup$
    – Dmitry Kerner
    Commented Nov 19, 2018 at 7:17

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