Levi Factors and Admissible Automorphisms

Abstract

Let $\\mathfrak {g}$\n be a complex simple Lie algebra. We consider subalgebras $\\mathfrak {m}$\n which are Levi factors of parabolic subalgebras of $\\mathfrak {g}$\n , or equivalently $\\mathfrak {m}$\n is the centralizer of its center. We introduced the notion of admissible systems on finite order $\\mathfrak {g}$\n -automorphisms 𝜃, and show that 𝜃 has admissible systems if and only if its fixed point set is a Levi factor. We then use the extended Dynkin diagrams to characterize such automorphisms, and look for automorphisms of minimal order.