Skew derivations of incidence algebras

TL;DR

This paper characterizes skew derivations of incidence algebras over locally finite posets, showing their structure and relating their quotient space to a cohomology group of the poset, extending classical derivation theory.

Contribution

It introduces a detailed description of $\

Findings

Decomposition of $\\varphi$-derivations similar to usual derivations.

Isomorphism between the quotient of $\\varphi$-derivations and a cohomology group.

Extension of derivation theory to automorphism-twisted derivations.

Abstract

In the first part of the paper we describe $\varphi$-derivations of the incidence algebra $I(X,K)$ of a locally finite poset $X$ over a field $K$, where $\varphi$ is an arbitrary automorphism of $I(X,K)$. We show that they admit decompositions similar to that of usual derivations of $I(X,K)$. In particular, the quotient of the space of $\varphi$-derivations of $I(X,K)$ by the subspace of inner $\varphi$-derivations of $I(X,K)$ is isomorphic to the first group of certain cohomology of $X$, which is developed in the second part of the paper.