On \(sl_2\)-triples in Lie algebras of reductive algebraic groups in positive characteristic

Pengelly, Rachel ORCID: 0000-0003-4640-4943 (2023). On \(sl_2\)-triples in Lie algebras of reductive algebraic groups in positive characteristic. University of Birmingham. Ph.D.

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Abstract

In this thesis we consider \(sl_2\)-triples in \(g\) = Lie(\(G\)), the Lie algebra of a connected reductive algebraic group over a field of positive characteristic \(p>2\).

We focus on good primes for \(G\) which are smaller than the coxeter number of \(G\), and determine to what extent the theorems of Jacobson--Morozov and Kostant hold in this setting. To do so, we determine the maximal \(G\)-stable closed subvariety \(V\) of the nilpotent cone \(N\) of \(g\) such that the \(G\)-orbits in \(V\) are in bijection with the \(G\)-orbits of \(sl_2\)-triples (\(e,h,f\)) with \(e,f\) \(\in\) \(V\).

We also determine the maximal \(G\)-stable closed subvariety \(V\) of the nilpotent cone \(N\) of \(g\) such that any subalgebra \(h\) = \( ⟨e,h,f⟩ \cong\) \(sl_2\) (\(k\)) with \(e\), \(f\) ∈ \(V\) is \(G\)-completely reducible

Type of Work:

Thesis (Doctorates > Ph.D.)

Award Type:

Doctorates > Ph.D.

Supervisor(s):