The Jacobson-Morozov Theorem and Complete Reduciblity of Lie subalgebras

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• Adam Thomas, University of Cambridge
• Wednesday 27 May 2015, 16:30-17:30
• MR12.

If you have a question about this talk, please contact David Stewart.

The well-known Jacobson-Morozov Theorem states that every nilpotent element of a complex semisimple Lie algebra $\mathfrak{g} = Lie(G)$ can be uniquely embedded in an $\mathfrak{sl}_2$-subalgebra, up to conjugacy by $G$. Much work has been done on extending this fundamental result to the modular case when $G$ is a reductive algebraic group over an algebraically closed field of characteristic $p > 0$. I will discuss recent joint work with David Stewart, proving that the theorem holds in the modular case precisely when $p$ is larger than $h(G)$, the Coxeter number of $G$. In doing so, we consider complete reduciblilty of subalgebras of $\mathfrak{g}$ in the sense of Serre/McNinch. For example, we prove that every $\mathfrak{sl}_2$-subalgebra of $\mathfrak{g}$ is completely reducible precisely when $ p > h(G)$.

This talk is part of the Algebra and Representation Theory Seminar series.

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