Automorphisms and derivations of Borel subalgebras and their nilradicals in Kac-Moody algebras

TL;DR

This paper characterizes automorphisms and derivations of Borel subalgebras and their nilradicals in Kac-Moody algebras over fields of characteristic zero, solving a longstanding conjecture and extending previous results.

Contribution

It provides a complete determination of automorphisms and derivations for these subalgebras in symmetrizable Kac-Moody algebras, generalizing prior work and resolving a 30-year-old conjecture.

Findings

Derivations of Borel subalgebras and nilradicals are fully characterized.

Automorphisms of these subalgebras are explicitly determined.

The results confirm the conjecture posed by R. V. Moody.

Abstract

In this paper, we determine derivations of Borel subalgebras and their derived subalgebras called nilradicals, in Kac-Moody algebras (and contragredient Lie algebras) over any field of characteristic 0; and we also determine automorphisms of those subalgebras in symmetrizable Kac-Moody algebras. The results solve a conjecture posed by R. V. Moody about 30 years ago which generalizes a result by B. Kostant and which is discussed by A. Fialowski using Lie algebra cohomology in case of affine type.