On Automorphisms of Enveloping Algebras

Abstract

Abstract Given an algebraic Lie algebra $\mathfrak{g}$ over $\mathbb{C}$, we canonically associate to it a Lie algebra $\mathcal{L}_{\infty }(\mathfrak{g})$ defined over $\mathbb{C}_{\infty }$, the reduction of $\mathbb{C}$ modulo the infinitely large prime, and show that for a class of Lie algebras, $\mathcal{L}_{\infty }(\mathfrak{g})$ 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 $\mathfrak{U}\mathfrak{g}$ to the group of Lie algebra automorphisms of $\mathfrak{g}$, such that its kernel does not contain a non-trivial semi-simple automorphism. As a corollary, we obtain that any finite subgroup of automorphisms of $\mathfrak{U}\mathfrak{g}$ is isomorphic to a subgroup of Lie algebra automorphisms of $\mathfrak{g}.$