Proper Lie automorphisms of incidence algebras

TL;DR

This paper investigates conditions under which all Lie automorphisms of incidence algebras of finite connected posets are proper, providing a sufficient condition and complete answers for specific classes of posets.

Contribution

It introduces a sufficient condition for proper Lie automorphisms and completely characterizes such automorphisms for certain classes of posets.

Findings

A sufficient condition involving an equivalence relation on maximal chains.

Complete characterization for crownless posets of length one.

Results for crowns and ordinal sums of two antichains.

Abstract

Let $X$ be a finite connected poset and $K$ a field. We study the question, when all Lie automorphisms of the incidence algebra $I(X,K)$ are proper. Without any restriction on the length of $X$ we find only a sufficient condition involving certain equivalence relation on the set of maximal chains of $X$. For some classes of posets of length one, such as finite connected crownless posets (i.e., without weak crown subposets), crowns and ordinal sums of two antichains we give a complete answer.