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Do morphisms of algebraic groups have any special properties? I am mainly interested in morphisms between algebraic groups preserving the group structure; but I am also interested in arbitrary morphisms between algebraic groups. For example,

What can be said about the morphism $x^n: G \rightarrow G$, $x$ goes into $x^n$: is it necessary proper? finite? when is it étale? what if $G$ is commutative but not necessarily an abelian variety?

In fact, I am rather looking for a reference which describes such properties in full generality: that is, between arbitrary (finite dimensional), possibly commutative, algebraic groups in arbitrary characteristic.

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    $\begingroup$ It is certainly not true that this map is proper for $\textbf{GL}_r$. The fiber over the identity is the same as the space of direct sum decompositions of an $r$-dimensional vector space indexed by the $\text{n}^{\text{th}}$ roots of unity. $\endgroup$ – Jason Starr Jul 7 '11 at 15:47
  • $\begingroup$ I think it's true that any identity-preserving morphism $G \to {\mathbb G}_m$, for $G$ reductive, is a homomorphism. Certainly this is easy to prove for ${\mathbb G}_m$ itself. $\endgroup$ – Allen Knutson Jul 7 '11 at 19:54
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These kinds of questions seem to be treated only in scattered papers over many years. Concerning power maps, see two related papers:

Chatterjee, Pralay, On the surjectivity of the power maps of semisimple algebraic groups. Math. Res. Lett. 10 (2003), no. 5-6, 625–633.

Steinberg, Robert, On power maps in algebraic groups. Math. Res. Lett. 10 (2003), no. 5-6, 621–624.

Both papers investigate when the map is surjective, as do two more recent papers by Chatterjee focusing on local fields. Steinberg's paper works in arbitrary characteristic. In both cases algebraic groups need to be connected.

I suspect there is no comprehensive paper of the kind wanted. The interaction between morphisms and algebraic group structure has been studied over many decades, but not definitively. As Allen remarks, an old theorem of Rosenlicht shows for example that for connected groups an identity-preserving morphism to the multiplicative group is a character. There are several simple modern proofs, one being readily accessible online:

Broughton, S. A., A note on characters of algebraic groups. Proc. Amer. Math. Soc. 89 (1983), no. 1, 39–40.

ADDED: In a somewhat complementary framework, there is also a lot of literature concerning group homomorphisms between linear algebraic groups which are not assumed to be morphisms; the most comprehensive and influential paper is the paper by Borel and Tits:

Borel, Armand; Tits, Jacques, Homomorphismes “abstraits” de groupes algebriques simples. Ann. of Math. (2) 97 (1973), 499–571.

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