Jacobson–Morozov theorem

The statement of Jacobson–Morozov relies on the following preliminary notions: an sl₂-triple in a semi-simple Lie algebra ${\displaystyle {\mathfrak {g}}}$ (throughout in this article, over a field of characteristic zero) is a homomorphism of Lie algebras ${\displaystyle {\mathfrak {sl}}_{2}\to {\mathfrak {g}}}$. Equivalently, it is a triple ${\displaystyle e,f,h}$ of elements in ${\displaystyle {\mathfrak {g}}}$ satisfying the relations

${\displaystyle [h,e]=2e,\quad [h,f]=-2f,\quad [e,f]=h.}$

An element ${\displaystyle x\in {\mathfrak {g}}}$ is called nilpotent, if the endomorphism ${\displaystyle [x,-]:{\mathfrak {g}}\to {\mathfrak {g}}}$ (known as the adjoint representation) is a nilpotent endomorphism. It is an elementary fact that for any sl₂-triple ${\displaystyle (e,f,h)}$, e must be nilpotent. The Jacobson–Morozov theorem states that, conversely, any nilpotent non-zero element ${\displaystyle e\in {\mathfrak {g}}}$ can be extended to an sl₂-triple.[1][2] For ${\displaystyle {\mathfrak {g}}={\mathfrak {sl}}_{n}}$, the sl₂-triples obtained in this way are made explicit in Chriss & Ginzburg (1997, 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 ${\displaystyle G_{a}}$ to a reductive group H factors through the embedding

${\displaystyle G_{a}\to SL_{2},x\mapsto \left({\begin{array}{cc}1&x\\0&1\end{array}}\right).}$

Furthermore, any two such factorizations

${\displaystyle SL_{2}\to H}$

are conjugate by a k-point of H.

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 ${\displaystyle G\to H}$ in both categories are taken up to conjugation by elements in ${\displaystyle H(k)}$, admits a left adjoint, the so-called pro-reductive envelope. This left adjoint sends the additive group ${\displaystyle G_{a}}$ to ${\displaystyle SL_{2}}$ (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 André & Kahn (2002, Theorem 19.3.1) by appealing to methods related to Tannakian categories and by O'Sullivan (2010) by more geometric methods.

• André, Yves; Kahn, Bruno (2002), "Nilpotence, radicaux et structures monoïdales", Rend. Semin. Mat. Univ. Padova, 108: 107–291, arXiv:math/0203273, Bibcode:2002math......3273A, MR 1956434
• Chriss, Neil; Ginzburg, Victor (1997), Representation theory and complex geometry, Birkhäuser, ISBN 0-8176-3792-3, MR 1433132
• Bourbaki, Nicolas (2007), Groupes et algèbres de Lie: Chapitres 7 et 8, Springer, ISBN 9783540339779
• Jacobson, Nathan (1935), "Rational methods in the theory of Lie algebras", Annals of Mathematics, Second Series, 36 (4): 875–881, doi:10.2307/1968593, JSTOR 1968593, MR 1503258
• Jacobson, Nathan (1979), Lie algebras (Republication of the 1962 original ed.), Dover Publications, Inc., New York, ISBN 0-486-63832-4
• Morozov, V. V. (1942), "On a nilpotent element in a semi-simple Lie algebra", C. R. (Doklady) Acad. Sci. URSS (N.S.), 36: 83–86, MR 0007750
• O'Sullivan, Peter (2010), "The generalised Jacobson-Morosov theorem", Memoirs of the American Mathematical Society, 207 (973), doi:10.1090/s0065-9266-10-00603-4, ISBN 978-0-8218-4895-1