Generic property and conjugacy classes of homogeneous Borel subalgebras of restricted Lie algebras

2014/04/04 by Shu, Bin · 2 citations
#17B 05 #17B 20 #17B 50 #FOS: Mathematics #Representation Theory (math.RT)

paper · doi:10.48550/arxiv.1404.1149

Abstract

Let (\mathfrakg,[p]) be a finite-dimensional restricted Lie algebra over an algebraically closed field \mathbbK of characteristic p>0, and G be the adjoint group of \mathfrakg. We say that \mathfrakg satisfying the \sl generic property if \mathfrakg admits generic tori introduced in \citeBFS. A Borel subalgebra (or Borel for short) of \mathfrakg is by definition a maximal solvable subalgebra containing a maximal torus of \mathfrakg, which is further called generic if additionally containing a generic torus. In this paper, we first settle a conjecture proposed by Premet in \citePr2 on regular Cartan subalgebras of restricted Lie algebras. We prove that the statement in the conjecture for a given \mathfrakg is valid if and only if it is the case when \mathfrakg satisfies the generic property. We then classify the conjugay classes of homogeneous Borel subalgebras of the restricted simple Lie algebras \mathfrakg=W(n) under G-conjugation when p>3, and present the representatives of these classes. Here W(n) is the so-called Jacobson-Witt algebra, by definition the derivation algebra of the truncated polynomial ring \mathbbK[T₁,⋯,T_n]\slash (T₁^p,⋯,T_n^p). We also describe the closed connected solvable subgroups of G associated with those representative Borel subalgebras.

Cited by

• Borel subalgebras of Lie algebras of vector fields
• Regular Borel subalgebras of the Lie Algebra of Aut(\mathbbA²)

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• Borel subalgebras of the Witt algebra W₁