Hostname: page-component-848d4c4894-75dct Total loading time: 0 Render date: 2024-06-02T03:50:36.629Z Has data issue: false hasContentIssue false
  • English
  • Français

Proper Lie automorphisms of incidence algebras

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

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
Get access Rights & Permissions [Opens in a new window]

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

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Baclawski, K., Automorphisms and derivations of incidence algebras, Proc. Amer. Math. Soc. 36(2) (1972), 351–356.CrossRefGoogle Scholar
Bollobás, B., Modern graph theory, vol. 184. Grad. Texts Math. (Springer, New York, NY, 1998).CrossRefGoogle Scholar
Brusamarello, R., Fornaroli, É. Z. and Santulo, E. A. Jr., Anti-automorphisms and involutions on (finitary) incidence algebras, Linear Multilinear Algebra 60(2) (2012), 181–188.CrossRefGoogle Scholar
Brusamarello, R., Fornaroli, É. Z. and Santulo, E. A. Jr., Classification of involutions on finitary incidence algebras, Int. J. Algebra Comput., 24(8) (2014), 1085–1098.CrossRefGoogle Scholar
Brusamarello, R., Fornaroli, É. Z. and Santulo, E. A. Jr., Multiplicative automorphisms of incidence algebras, Commun. Algebra 43(2) (2015), 726–736.CrossRefGoogle Scholar
Brusamarello, R. and Lewis, D. W., Automorphisms and involutions on incidence algebras, Linear Multilinear Algebra 59(11) (2011), 1247–1267.CrossRefGoogle Scholar
Cao, Y., Automorphisms of certain Lie algebras of upper triangular matrices over a commutative ring, J. Algebra 189(2) (1997), 506–513.Google Scholar
Cecil, A. J., Lie isomorphisms of triangular and block-triangular matrix algebras over commutative rings, Master’s thesis, University of Victoria, 2016.Google Scholar
Doković, D. Ž., Automorphisms of the Lie algebra of upper triangular matrices over a connected commutative ring, J. Algebra 170(1) (1994), 101–110.CrossRefGoogle Scholar
Dokuchaev, M. and Novikov, B., On colimits over arbitrary posets, Glasg. Math. J. 58(1) (2016), 219–228.CrossRefGoogle Scholar
Dräxler, P., Completely separating algebras, J. Algebra 165(3) (1994), 550–565.CrossRefGoogle Scholar
Drozd, Y. and Kolesnik, P., Automorphisms of incidence algebras, Comm. Algebra 35(12) (2007), 3851–3854.CrossRefGoogle Scholar
Fornaroli, É. Z., Khrypchenko, M. and Santulo, E. A. Jr, Lie automorphisms of incidence algebras, To appear in Proc. Amer. Math. Soc. (2020). (arXiv:2012.06661v3). DOI: 10.1090/proc/15786.CrossRefGoogle Scholar
Hua, L. K., A theorem on matrices over a sfield and its applications, J. Chinese Math. Soc. (N.S.) 1 (1951), 110–163.Google Scholar
Khripchenko, N. S., Automorphisms of finitary incidence rings, Algebra and Discrete Math. 9(2) (2010), 78–97.Google Scholar
Martindale, W. S. 3rd, Lie isomorphisms of primitive rings, Proc. Amer. Math. Soc. 14 (1963), 909–916.CrossRefGoogle Scholar
Martindale, W. S. 3rd, Lie isomorphisms of prime rings, Trans. Amer. Math. Soc. 142 (1969), 437–455.CrossRefGoogle Scholar
Martindale, W. S. 3rd, Lie isomorphisms of simple rings, J. Lond. Math. Soc. 44 (1969), 213–221.CrossRefGoogle Scholar
Rival, I., A fixed point theorem for finite partially orderes sets, J. Comb. Theory, Ser. A 21 (1976), 309–318.CrossRefGoogle Scholar
Rota, G.-C., On the foundations of combinatorial theory. I. Theory of Möbius functions, Z. Wahrscheinlichkeitstheorie und Verw. Gebiete 2(4) (1964), 340–368.Google Scholar
Spiegel, E., On the automorphisms of incidence algebras, J. Algebra 239(2) (2001), 615–623.CrossRefGoogle Scholar
Spiegel, E. and O’Donnell, C. J., Incidence Algebras (Marcel Dekker, New York, NY, 1997).Google Scholar
Stanley, R., Structure of incidence algebras and their automorphism groups, Bull. Am. Math. Soc. 76 (1970), 1236–1239.CrossRefGoogle Scholar
Stanley, R. P., Enumerative combinatorics. Vol. 1, vol. 49. Camb. Stud. Adv. Math. (Cambridge University Press, Cambridge, 1997).CrossRefGoogle Scholar