# Jacobson-Morozov on the algebraic group level

Let $k$ be a field of characteristic 0 and $G$ a connected reductive algebraic group defined over $k$. Let $g \in G(k)$ be a unipotent element. Is it true that there is a homomorphism $\varphi: SL_2(k) \to G(k)$ mapping some unipotent element $u \in SL_2(k)$ to $g$?

Background: There is some nilpotent element $n \in Lie\, G$ which corresponds to $u$. If $Lie\, G$ is semisimple, the Jacobson-Morozov lemma states that $n$ is part of a $sl_2$-triple, so ''integrating'' would yield the desired homomorphism. (Apparently, Jacobson-Morozov holds more generally for completely reducible Lie subalgebras of $gl(V)$, though I don't have a precise reference.)

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You mean $\varphi$ is hom. ${\rm{SL}}_ 2 \rightarrow G$ as $k$-gps, not merely homomorphism of gps of $k$-points. Zariski closure $U$ of $\langle g rangle$ is comm. and unip., hence vector gp since in char 0. Then get $\mathbf{G}_ a$ containing $g$ and must be $U$ if $g \ne 1$ (as we may and do assume). Apply J-M to $\mathfrak{u} = {\rm{Lie}}(U)$ to get $\mathfrak{sl}_ 2$ in $\mathfrak{g}$ containing $\mathfrak{u}$. By Cor 7.9 in Borel's alg. gp. book, in char. 0 perfect Lie subalg "integrates" to closed $k$-subgp. That's ${\rm{SL}}_ 2$ or ${\rm{PGL}}_ 2$ containing $U$, and does the job. –  BCnrd Apr 22 '10 at 15:15

## 1 Answer

Much of the literature in characteristic 0 is older and may not immediately fit the exact format of your question. But here is a sample:

N. Jacobson's 1962 book Lie Algebras (later reprinted by Dover) discusses in II.5 (and II.11) the Lie algebra structure of a completely reducible linear Lie algebra in characteristic 0. His Theorem 8 shows that the Lie algebra decomposes into its center plus a semisimple ideal. (In the parallel algebraic group setting, the Lie algebra of a completely reducible linear algebraic group in characteristic 0 has this form.) The semisimple or just simple case is really the crucial one for Jacobson-Morozov theory.

In the 1968-69 IAS seminar volume published as Springer LN 131 in 1970, look at section III.4 in Conjugacy classes by T.A. Springer and R. Steinberg; here there are also adaptations to prime characteristic.

R.W. Carter's 1985 book Finite Groups of Lie Type (Chapter 5) has a nice treatment, though he usually works with simple algebraic groups.

There is also a Lie algebra treatment in section 11 of Bourbaki's Chap. 8 in the Lie groups series, supplemented by interesting exercises.

In characteristic 0 the exponential map works well to pass from the Lie algebra to the group, but in characteristic $p$ the Jacobson-Morozov argument only works for large enough $p$. For refinements involving the groups when $p$ is small, there are substantial papers by G. McNinch and D. Testerman in the past decade or so. At any rate, the case $g=1$ or $n=0$ of your question is trivial and can be left aside.

As BCnrd observes, you have to be careful to specify your maps to be algebraic group homomorphisms. In characteristic 0, working with an algebraically closed field isn't so important, but in general you have to treat points of the group over a field with care. (There is a careful study by Borel and Tits of the way abstract group homomorphisms relate to algebraic group morphisms, but you don't want to get into that here.)

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