Algebraic groups “generated” by a Lie algebra element

Here is a definition which I invented and which I would like to understand better.

Let $A$ be a complex affine algebraic group. Let $X \in \mathfrak g$ be an element in its Lie algebra. We say that $X$ generates $A$ if there is no proper subgroup of $A$ whose Lie algebra contains $X$.

Is this a standard notion? Are the following two facts true?

(i) If $A$ is generated by $X$, then $A$ is abelian.

(ii) If $V$ is a representation of $A$ and $X$ acts by $0$ on $V$, then $A$ acts by the identity on $V$.

The main example that I have in mind is as follows. Let $G$ be a reductive group and let $X$ be a regular element. Let $A$ be the centralizer of $X$ in $G$. Then $A$ is generated by $X$.

Even more, I would like to know if this works in families. Take $A$ to be the group scheme of regular centralizers, viewed as a group scheme over $\mathfrak t / W$. Then I would like to know that $A$ is generated by the universal section $X$ and I would like to know that the conclusion of (ii) holds in this case.

So to ask some specific questions:

1. Does this notion of "generation" exist in the literature?

2. Is (ii) above always true?

3. Can I apply this to the group of regular centralizers?

Update: In light of Jim's answer below, I realize that the centralizer of a principal nilpotent is not generated by the principal nilpotent. (I was just being stupid.) To make this question a little more interesting, I offer up one more claim:

1. A complex algebraic group $A$ is generated by an element of its Lie algebra if and only if $A = T \rtimes V$ where $T$ is a torus, $V$ is a vector space (viewed as a group under addition) such that $T$ acts linearly on $V$ with one-dimensional weight spaces.

For example, this means that any torus is generated by an element of its Lie algebra (the case $V = 0$) and that the only unipotent group generated by an element of its Lie algebra is $\mathbb G_a$ (the case $T = \{1 \}$).

There are a number of questions asked, sometimes in approximate language, so I'm not sure what the main focus is. To begin with, the crucial thing here is to work over an algebraically closed field of characteristic 0, where (connected) affine/linear algebraic groups and their Lie algebras coordinate well. In particular, their finite dimensional representations behave well. (The complex field has no direct bearing on what you ask.)

Chevalley initiated the study of such matters, developing the analogy with Lie groups, but his 1950s language is now outdated. The best semi-modern starting point is Section 7 of Borel's GTM 126 Linear Algebraic Groups, which incorporates material from his earlier lecture notes. Here he works out pretty explicitly the structure of closed connected subgroups of an affine algebraic group "generated" by subsets of the group: the definitions make sense in arbitrary characteristic, but the analogous notions for Lie algebras require characteristic 0. Given an element $X$ of the Lie algebra, one gets a smallest closed subgroup of $G$ whose Lie algebra contains $X$. In 7.3 he works out the matrix case explicitly, which depends on the Jordan decomposition.

Further aspects of all this are worked out in two papers by Borel and Springer, starting with the proceedings of the 1965 Boulder AMS Summer Institute. Steinberg made the situation more explicit (in arbitrary characteristic) for regular elements of reductive groups: those whose centralizers have minimal dimension, necessarily equal to the rank of $G$. In particular, these centralizers always turn out to be abelian. But the smallest algebraic group whose Lie algebra contains a regular nilpotent element of $\mathfrak{g}$ is only 1-dimensional. So I'm not sure what it would mean for such an element to "generate" its centralizer in the group.

Concerning your last question on the "group of regular centralizers", I'm not sure what this notion means. Please clarify your question(s), if possible.

ADDED: I still think you need to work in the algebraic situation (characteristic 0), where you can exploit the Jordan decomposition as Borel does in his updated sketch of Chevalley's older results. Your added question 4 is out of focus, since the Jordan decomposition gives a direct product (or direct sum of Lie algebras) here, possibly involving just a unipotent group of dimension 1 for a nonzero nilpotent element in the Lie algebra. [Note also that your semi-direct product symbol seems to be backwards. LaTeX gives two choices $\ltimes$ and $\rtimes$.] For the semisimple part you can get tori of varying dimensions, as Borel explains just in the general linear case. (So you have to embed your affine algebraic group in a general linear group.)

To sum up, your notion of a Lie algebra element "generating" an algebraic group or its Lie algebra basically originates in the older notions developed by Chevalley and then Borel, which are more precise than your version.

(i) is true: A is equal to the stabiliser of X in the adjoint representation (otherwise this would be a smaller subgroup whose Lie algebra contains X). Also A is connected. Then X is central in Lie(A). Therefore it lies in the Lie algebra of the centre of A. By the generation hypothesis, A is equal to its centre, hence abelian.

(ii) is also true: A representation of A on V is a map pi:A-->GL(V). Write Lie(pi) for the map between their Lie algebras. Then Lie(ker(pi))=ker(Lie(pi)). Since X is in ker(Lie(pi)), ker(pi) is a subgroup of A whose Lie algebra contains X. By the generation hypothesis, ker(pi)=A, hence A acts trivially on V.

For question 3, to check whether a representation of a group scheme over an interesting base (e.g. t/W) is trivial, it suffices to check triviality at each closed point. Then it looks as though you're back in the case of a group over the complex numbers, where the action has just been shown to be trivial.

• Thanks. What about my last question? Can I apply this to the group of regular centralizers? – Joel Kamnitzer May 29 '15 at 9:21

In addition to Jim's comments, let me point out that a relevant term here is the notion of replica of an endomorphism, introduced by Chevalley. If you google this word, you will find many references to it. A nice treatment appears in Serre's book "Lie algebras and Lie groups", Section V.6. In particular, Theorem V.6.8 of this book seems to be closely related to Question (ii) above.

There is plenty of example where this notion of group generated by elements of Lie algebra appears. For instance, if $X$ is semisimple you may look for smallest torus whose lie algebra contains $X$. Again, if you google search the above phrase you see several references (including Serre's book). It is clear that this notion has been used extensively since, at least, Chevalley.

A recent example is the article of Frenkel-Gross. In Section 13, they show that inside a simple group $G$ of rank $n$, the torus generated by the element $N+E$ has dimension $\phi(h)$. Here, $N=e_1+...+e_n$ is a principal nilpotent element and $E$ is the basis vector for the lowest root. By Kostant's Theorem, $N+E$ is regular semisimple. Finally $h$ is the Coxeter number and $\phi(h)$ is the number of integers less than $h$ which are prime to $h$. (In particular, since the generated torus is obvious contained in a maximal torus, this argument provides a cute proof that $\phi(h)\leq n=\mathrm{rank}$ in any root system.)

Also, both the example of $X$ being the principal nilpotent element and $X=N+E$ seem to show that in general the generated group is not the stabiliser in the adjoint representation. If I understand the situation correctly, the stabiliser always has dimension $\geq$ rank, whereas we know that the dimension of the generated group can be smaller. So I'm confused about Peter's remark.