Potent preservers of incidence algebras

TL;DR

This paper characterizes the structure of bijective linear maps that preserve idempotents, tripotents, and k-potents in incidence algebras of finite connected posets, revealing their automorphism, anti-automorphism, and Lie automorphism nature depending on the field's characteristic.

Contribution

It provides a comprehensive description of linear preservers of algebraic elements in incidence algebras, including automorphisms, anti-automorphisms, and Lie automorphisms, based on the field's characteristic.

Findings

Preservers are automorphisms or anti-automorphisms when char(F) ≠ 2.

In characteristic 2, preservers are Lie automorphisms.

For F=Z_2, preservers combine shift maps and Lie automorphisms.

Abstract

Let $X$ be a finite connected poset, $F$ a field and $I(X,F)$ the incidence algebra of $X$ over $F$. We describe the bijective linear idempotent preservers $\varphi:I(X,F)\to I(X,F)$. Namely, we prove that, whenever $\mathrm{char}(F)\ne 2$, $\varphi$ is either an automorphism or an anti-automorphism of $I(X,F)$. If $\mathrm{char}(F)=2$ and $|F|>2$, then $\varphi$ is a (in general, non-proper) Lie automorphism of $I(X,F)$. Finally, if $F=\mathbb{Z}_2$, then $\varphi$ is the composition of a bijective shift map and a Lie automorphism of $I(X,F)$. Under certain restrictions on the characteristic of $F$ we also obtain descriptions of the bijective linear maps which preserve tripotents and, more generally, $k$-potents of $I(X,F)$ for $k\ge 3$.