Jacobson–Morozov theorem

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 {{harvnb|Jacobson|1951}}, {{harvnb|Morozov|1942}}.

Statement

The statement of Jacobson–Morozov relies on the following preliminary notions: an sl₂-triple in a semi-simple Lie algebra $\mathfrak g$ (throughout in this article, over a field of characteristic zero) is a homomorphism of Lie algebras $\mathfrak{sl}_2 \to \mathfrak g$ . Equivalently, it is a triple $e, f, h$ of elements in $\mathfrak g$ satisfying the relations

: $[h,e] = 2e, \quad [h,f] = -2f, \quad [e,f] = h.$

An element $x \in \mathfrak g$ is called nilpotent, if the endomorphism $[x, -] : \mathfrak g \to \mathfrak g$ (known as the adjoint representation) is a nilpotent endomorphism. It is an elementary fact that for any sl₂-triple $(e, f, h)$ , e must be nilpotent. The Jacobson–Morozov theorem states that, conversely, any nilpotent non-zero element $e \in \mathfrak g$ can be extended to an sl₂-triple.{{harvtxt|Bourbaki|2007|loc=Ch. VIII, §11, Prop. 2}}{{harvtxt|Jacobson|1979|loc=Ch. III, §11, Theorem 17}} For $\mathfrak g = \mathfrak{sl}_n$ , the sl₂-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 groups (again over a field k of characteristic zero): any morphism (of algebraic groups) from the additive group $G_a$ to a reductive group H factors through the embedding

: $G_a \to SL_2, x \mapsto \left ( \begin{array}{cc} 1 & x \\ 0 & 1 \end{array} \right ).$

Furthermore, any two such factorizations

: $SL_2 \to H$

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 $G \to H$ in both categories are taken up to conjugation by elements in $H(k)$ , admits a left adjoint, the so-called pro-reductive envelope. This left adjoint sends the additive group $G_a$ to $SL_2$ (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 {{harvtxt|André|Kahn|2002|loc=Theorem 19.3.1}} by appealing to methods related to Tannakian categories and by {{harvtxt|O'Sullivan|2010}} by more geometric methods.

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}}

Category:Lie algebras

Category:Algebraic groups

Category:Theorems in algebra