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.
The statement of Jacobson–Morozov relies on the following preliminary notions: an sl2-triple in a semi-simple Lie algebra
(throughout in this article, over a field of characteristic zero) is a homomorphism of Lie algebras
(known as the adjoint representation) is a nilpotent endomorphism.
The Jacobson–Morozov theorem states that, conversely, any nilpotent non-zero element
, the sl2-triples obtained in this way are made explicit in Chriss & Ginzburg (1997, p. 184).
to a reductive group H factors through the embedding Furthermore, any two such factorizations 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
, admits a left adjoint, the so-called pro-reductive envelope.
This left adjoint sends the additive group
(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.