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 Jacobson 1951, Morozov 1942.

Statement

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.

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 ${\displaystyle G\to H}$ 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.

References

1. Bourbaki (2007, Ch. VIII, §11, Prop. 2)
2. Jacobson (1979, Ch. III, §11, Theorem 17)

• 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 (1951), "Completely reducible Lie algebras of linear transformations", Proceedings of the American Mathematical Society, 2: 105–113, doi:10.1090/S0002-9939-1951-0049882-5, MR 0049882
• 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, New Series, 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