Simple Lie algebras of characteristic 𝑝 with dependent roots

Abstract

Abstract.We investigate finite dimensional simple Lie algebras over an algebraically closed field F of characteristic p > 7 having a Cartan subalgebra H whose roots are dependent over F . We show that H must be one-dimensional or for some root a relative to H there is a 1-section L such that the core of L is a simple Lie algebra of Cartan type H(2 : m : 4>) or W(\ : n) for some n > 1 . The results we obtain have applications to studying the local behavior of simple Lie algebras and to classifying simple Lie algebras which have a Cartan subalgebra of dimension less than p -2 .

Gems from the Work of Georgia Benkart

“…Benkart and Osborn [BO84] classified the finitedimensional simple Lie algebras of characteristic 𝑝 > 7 with a one-dimensional Cartan subalgebra, showing that they are either 𝔰𝔩(2) or Albert-Zassenhaus Lie algebras (the algebras 𝑊(1, 𝑛) and a family of Hamiltonian Lie algebras). Their paper [BO90] studied the subalgebra 𝐿 (𝛼) = 𝐿 0 ⊕ 𝐿 𝛼 ⊕ 𝐿 2𝛼 ⊕ ⋯ ⊕ 𝐿 (𝑝−1)𝛼 of a finite-dimensional simple Lie algebra 𝐿 determined by a root 𝛼. Modulo the radical, these one-sections 𝐿 (𝛼) are isomorphic to either 𝔰𝔩(2), 𝑊(1, 1), or to a subalgebra of 𝐻(2, 1) containing 𝐻(2, 1) (2) .…”