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{{Technical|date=December 2019}}
In mathematics, the '''Jacobson&ndash;Morozov theorem''' is the assertion that [[nilpotent]] elements in a semi-simple [[Lie algebra]] can be extended to [[sl2-triple|sl<sub>2</sub>-triples]]. The theorem is named after {{harvnb|Jacobson|1951}}, {{harvnb|Morozov|1942}}.

==Statement==

The statement of Jacobson&ndash;Morozov relies on the following preliminary notions: an sl<sub>2</sub>-triple in a [[semi-simple Lie algebra]] <math>\mathfrak g</math> (throughout in this article, over a field of [[characteristic zero]]) is a [[homomorphism]] of Lie algebras <math>\mathfrak{sl}_2 \to \mathfrak g</math>. Equivalently, it is a triple <math>e, f, h</math> of elements in <math>\mathfrak g</math> satisfying the relations 
:<math>[h,e] = 2e, \quad [h,f] = -2f, \quad [e,f] = h. </math>
An element <math>x \in \mathfrak g</math> is called nilpotent, if the [[endomorphism]] <math>[x, -] : \mathfrak g \to \mathfrak g</math> (known as the [[adjoint representation]]) is a [[nilpotent endomorphism]]. It is an elementary fact that for any sl<sub>2</sub>-triple <math>(e, f, h)</math>, ''e'' must be nilpotent. The Jacobson&ndash;Morozov theorem states that, conversely, any nilpotent non-zero element <math>e \in \mathfrak g</math> can be extended to an sl<sub>2</sub>-triple.<ref>{{harvtxt|Bourbaki|2007|loc=Ch. VIII, §11, Prop. 2}}</ref><ref>{{harvtxt|Jacobson|1979|loc=Ch. III, §11, Theorem 17}}</ref> For <math>\mathfrak g = \mathfrak{sl}_n</math>, the sl<sub>2</sub>-triples obtained in this way are made explicit in {{harvtxt|Chriss|Ginzburg|1997|loc=p. 184}}.

The theorem can also be stated for [[linear algebraic group]]s (again over a field ''k'' of characteristic zero): any morphism (of algebraic groups) from the [[additive group]] <math>G_a</math> to a [[reductive group]] ''H'' factors through the embedding 
:<math>G_a \to SL_2, x \mapsto \left ( \begin{array}{cc} 1 & x \\ 0 & 1 \end{array} \right ).</math>
Furthermore, any two such factorizations
:<math>SL_2 \to H</math>
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 <math>G \to H</math> in both categories are taken up to conjugation by elements in <math>H(k)</math>, admits a [[left adjoint]], the so-called pro-reductive envelope. This left adjoint sends the additive group <math>G_a</math> to <math>SL_2</math> (which happens to be semi-simple, as opposed to pro-reductive), thereby recovering the above form of Jacobson&ndash;Morozov.
This generalized Jacobson&ndash;Morozov theorem was proven by {{harvtxt|André|Kahn|2002|loc=Theorem 19.3.1}} by appealing to methods related to [[Tannakian category|Tannakian categories]] and by {{harvtxt|O'Sullivan|2010}} by more geometric methods.

==References==

<references/>

*{{Citation|last1=André|first1=Yves|last2=Kahn|first2=Bruno|title=Nilpotence, radicaux et structures monoïdales|journal=Rend. Semin. Mat. Univ. Padova|volume=108|year=2002|pages=107–291|mr=1956434|arxiv=math/0203273|bibcode=2002math......3273A}}
*{{Citation|last1=Chriss|first1=Neil|last2=Ginzburg|first2=Victor|title=Representation theory and complex geometry|publisher=Birkhäuser|year=1997|isbn=0-8176-3792-3|mr=1433132}}		
*{{Citation|first=Nicolas|last=Bourbaki|title=Groupes et algèbres de Lie: Chapitres 7 et 8|publisher=Springer|year=2007|isbn=9783540339779}}
*{{Citation|last=Jacobson|first=Nathan|title=Rational methods in the theory of Lie algebras|journal=Annals of Mathematics |series=Second Series|volume=36|year=1935|issue=4|pages=875–881|mr=1503258|doi=10.2307/1968593|jstor=1968593}}
*{{Citation|last=Jacobson|first=Nathan|title=Completely reducible Lie algebras of linear transformations|journal=Proceedings of the American Mathematical Society  |year=1951|volume=2 |pages=105–113|doi=10.1090/S0002-9939-1951-0049882-5 |mr=0049882|doi-access=free}}
*{{Citation|last=Jacobson|first=Nathan|author-link=Nathan Jacobson|title=Lie algebras|edition=Republication of the 1962 original|publisher=Dover Publications, Inc., New York|year=1979|isbn=0-486-63832-4}}
*{{Citation|last=Morozov|first=V. V.|title=On a nilpotent element in a semi-simple Lie algebra|journal=C. R. (Doklady) Acad. Sci. URSS |series=New Series|volume=36|year=1942|pages=83–86|mr=0007750}}
*{{Citation|last=O'Sullivan|first=Peter|title=The generalised Jacobson-Morosov theorem|journal=Memoirs of the American Mathematical Society|volume=207|issue=973|year=2010|isbn=978-0-8218-4895-1|doi=10.1090/s0065-9266-10-00603-4}}

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[[Category:Lie algebras]]
[[Category:Algebraic groups]]