nLab Jacobson-Morozov theorem

Statement

If $\mathfrak{g}$ is a semisimple Lie algebra over ground field the complex numbers, then for every nilpotent element $x \in \mathfrak{g}$ (i.e. $\underset{n \in \mathbb{N}}{\exists} [x,-]^n = 0$) there is as homomorphism of Lie algebras

$$\mathfrak{sl}(2,\mathbb{C}) \longrightarrow \mathfrak{g}$$

from sl(2), such that $x$ is the image of a nilpotent element of $\mathfrak{sl}(2,\mathbb{C})$.

References

Named after:

• V. V. Morozov: On a nilpotent element in a semi-simple Lie algebra, C. R. (Doklady) Acad. Sci. URSS, New Series 36 (1942) 83–86

• Nathan Jacobson: Completely reducible Lie algebras of linear transformations, Proc. Amer. Math. Soc. 2 (1951) 105-113 [doi:10.1090/S0002-9939-1951-0049882-5]

Further early discussion:

• Bertram Kostant, The principal three-dimensional subgroup and the Betti numbers of a complex simple Lie group, Amer. J. Math. 81 (1959) 973–1032 [jstor:2372999]

Lecture notes:

• Ana Balibanu, The Jacobson-Morozov theorem, Section 5 of“ Geometry of semisimple Lie algebras (pdf)

See also

• Wikipedia: Jacobson-Morozov theorem

• Wikipedia, sl2-triple