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Jacobson–Morozov theorem

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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

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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 . Equivalently, it is a triple of elements in satisfying the relations

An element is called nilpotent, if the endomorphism (known as the adjoint representation) is a nilpotent endomorphism. It is an elementary fact that for any sl2-triple , e must be nilpotent. The Jacobson–Morozov theorem states that, conversely, any nilpotent non-zero element can be extended to an sl2-triple.[1][2] For , 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 to a reductive group H factors through the embedding

Furthermore, any two such factorizations

are conjugate by a k-point of H.

Generalization

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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 in both categories are taken up to conjugation by elements in , admits a left adjoint, the so-called pro-reductive envelope. This left adjoint sends the additive group to (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

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  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