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Proper Lie automorphisms of incidence algebras

Part of: Rings and algebras with additional structure Ordered sets Rings and algebras arising under various constructions Lie algebras and Lie superalgebras Graph theory

Published online by Cambridge University Press:  07 February 2022

Érica Z. Fornaroli ,
Mykola Khrypchenko  and
Ednei A. Santulo Jr.
Érica Z. Fornaroli
Affiliation:
Departamento de Matemática, Universidade Estadual de Maringá, Maringá, PR, CEP: 87020–900, Brazilezancanella@uem.br, easjunior@uem.br
Mykola Khrypchenko
Affiliation:
Departamento de Matemática, Universidade Federal de Santa Catarina, Campus Reitor João David Ferreira Lima, Florianópolis, SC, CEP: 88040–900, Brazilnskhripchenko@gmail.com
Ednei A. Santulo Jr.
Affiliation:
Departamento de Matemática, Universidade Estadual de Maringá, Maringá, PR, CEP: 87020–900, Brazilezancanella@uem.br, easjunior@uem.br
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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 anti-chains, we give a complete answer.

Keywords

Lie automorphismproper Lie automorphismincidence algebramaximal chaincrown

MSC classification

Primary: 16S50: Endomorphism rings; matrix rings 17B60: Lie (super)algebras associated with other structures (associative, Jordan, etc.) 17B40: Automorphisms, derivations, other operators 06A07: Combinatorics of partially ordered sets
Secondary: 16W10: Rings with involution; Lie, Jordan and other nonassociative structures 05C38: Paths and cycles
Type
Research Article
Information
Glasgow Mathematical Journal , Volume 64 , Issue 3 , September 2022 , pp. 702 - 715
Copyright
© The Author(s), 2022. Published by Cambridge University Press on behalf of Glasgow Mathematical Journal Trust

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