On automorphisms of enveloping algebras

Abstract

Given an algebraic Lie algebra $\mathfrak{g}$ over $\mathbb{C}$, we canonically associate to it a Lie algebra ${\mathfrak{g}}_{\infty }$ defined over ${\mathbb{C}}_{\infty }$-the reduction of $\mathbb{C}$ mod infinitely large prime, and show that for a class of Lie algebras ${\mathfrak{g}}_{\infty }$ is an invariant of the derived category of $\mathfrak{g}$-modules. We give two applications of this construction. First, we show that the bounded derived category of $\mathfrak{g}$-modules determines algebra $\mathfrak{g}$ for a class of Lie algebras. Second, given a semi-simple Lie algebra $\mathfrak{g}$ over $\mathbb{C}$, we construct a canonical homomorphism from the group of automorphisms of the enveloping algebra $\mathfrak{Ug}$ to the group of Lie algebra automorphisms of $\mathfrak{g}$, such that its kernel does not contain a nontrivial semi-simple automorphism. As a corollary we obtain that any finite subgroup of automorphisms of $\mathfrak{Ug}$ isomorphic to a subgroup of Lie algebra automorphisms of $\mathfrak{g}.$

12 pages, to appear in the IMRN

References in corpus (1)

• Localization of modules for a semisimple Lie algebra in prime characteristic