Lie-orthogonal operators

TL;DR

This paper investigates Lie-orthogonal operators on finite-dimensional Lie algebras, revealing their invariance properties, characterizing them for simple, semi-simple, and reductive algebras, and computing their sets for specific classes like Heisenberg algebras.

Contribution

It provides a complete description of Lie-orthogonal operators on simple, semi-simple, and reductive Lie algebras, and explores their structure on various specific classes.

Findings

Lie-orthogonal operators preserve the center, radical, and central series components.

On simple Lie algebras, Lie-orthogonal operators are only trivial.

The group of Lie-orthogonal operators on a Heisenberg algebra is isomorphic to a symplectic group.

Abstract

Basic properties of Lie-orthogonal operators on a finite-dimensional Lie algebra are studied. In particular, the center, the radical and the components of the ascending central series prove to be invariant with respect to any Lie-orthogonal operator. Over an algebraically closed field of characteristic 0, only solvable Lie algebras with solvability degree not greater than two admit Lie-orthogonal operators whose all eigenvalues differ from 1 and -1. The main result of the paper is that Lie-orthogonal operators on a simple Lie algebra are exhausted by the trivial ones. This allows us to give the complete description of Lie-orthogonal operators for semi-simple and reductive algebras, as well as a preliminary description of Lie-orthogonal operators on Lie algebras with nontrivial Levi-Mal'tsev decomposition. The sets of Lie-orthogonal operators of some classes of Lie algebras (Heisenberg algebras, almost Abelian algebras, etc.) are directly computed. In particular, it appears that the group formed by the equivalence classes of Lie-orthogonal operators on a Heisenberg algebra is isomorphic to the standard symplectic group of an appropriate dimension.