Jacobson–Morozov theorem
Template:Technical In mathematics, the Jacobson–Morozov theorem is the assertion that nilpotent elements in a semi-simple Lie algebra can be extended to sl2-triples. The theorem is named after Template:Harvnb, Template:Harvnb.
Statement
The statement of Jacobson–Morozov relies on the following preliminary notions: an sl2-triple in a semi-simple Lie algebra (throughout in this article, over a field of characteristic zero) is a homomorphism of Lie algebras . Equivalently, it is a triple of elements in satisfying the relations
An element is called nilpotent, if the endomorphism (known as the adjoint representation) is a nilpotent endomorphism. It is an elementary fact that for any sl2-triple , e must be nilpotent. The Jacobson–Morozov theorem states that, conversely, any nilpotent non-zero element can be extended to an sl2-triple.[1][2] For , the sl2-triples obtained in this way are made explicit in Template:Harvtxt.
The theorem can also be stated for linear algebraic groups (again over a field k of characteristic zero): any morphism (of algebraic groups) from the additive group to a reductive group H factors through the embedding
Furthermore, any two such factorizations
are conjugate by a k-point of H.
Generalization
A far-reaching generalization of the theorem as formulated above can be stated as follows: the inclusion of pro-reductive groups into all linear algebraic groups, where morphisms in both categories are taken up to conjugation by elements in , admits a left adjoint, the so-called pro-reductive envelope. This left adjoint sends the additive group to (which happens to be semi-simple, as opposed to pro-reductive), thereby recovering the above form of Jacobson–Morozov. This generalized Jacobson–Morozov theorem was proven by Template:Harvtxt by appealing to methods related to Tannakian categories and by Template:Harvtxt by more geometric methods.