Jacobson–Morozov theorem

Jacobson–Morozov theorem This article may be too technical for most readers to understand. Please help improve it to make it understandable to non-experts, without removing the technical details. (December 2019) (Learn how and when to remove this template message) 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 xin {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 ein {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},xmapsto left({begin{array}{cc}1&x\0&1end{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 Gto 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 ^ Bourbaki (2007, Ch. VIII, §11, Prop. 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 Categories: Lie algebrasAlgebraic groups