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.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
Notes
Notice that it also follows from Corollary 3.13 that \(H^1_{(\sigma ,\lambda )}(X,K)\cong H^1_{(\zeta ,\lambda )}(X,K)\) for any multiplicative \(\sigma\).
In fact, \(H^1_{(\zeta ,\textrm{id})}(X,K)\cong K\), because \(\tau \in Z^1_{(\zeta ,\textrm{id})}(X,K)\) belongs to \(B^1_{(\zeta ,\textrm{id})}(X,K)\) if and only if \(\tau (1,5)-\tau (2,5)+\tau (2,6)-\tau (3,6)+\tau (3,7)-\tau (4,7)+\tau (4,8)-\tau (1,8)=0\).
References
Baclawski, K.: Automorphisms and derivations of incidence algebras. Proc. Am. Math. Soc. 36(2), 351–356 (1972)
Benkovič, D.: Jordan \(\sigma\)-derivations of triangular algebras. Linear Multilinear Algebra 64(2), 143–155 (2016)
Benkovič, D.: Lie \(\sigma\)-derivations of triangular algebras. Linear Multilinear Algebra 70(15), 2966–2983 (2022)
Brešar, M.: On generalized biderivations and related maps. J. Algebra 172(3), 764–786 (1995)
Brusamarello, R., Fornaroli, E.Z., Santulo, E.A., Jr.: Multiplicative automorphisms of incidence algebras. Commun. Algebra 43, 726–736 (2015)
Fornaroli, É.Z., Pezzott, R.E.M.: Additive derivations of incidence algebras. Commun. Algebra 49(4), 1816–1828 (2021)
Khripchenko, N.S.: Derivations of finitary incidence rings. Commun. Algebra 40(7), 2503–2522 (2012)
Khrypchenko, M.: Jordan derivations of finitary incidence rings. Linear Multilinear Algebra 64(10), 2104–2118 (2016)
Loday, J.-L.: Cyclic homology, vol. 301 of Grundlehren der mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences]. Springer, Berlin, 1992. Appendix E by María O. Ronco
Martín González, C., Repka, J., Sánchez-Ortega, J.: Automorphisms, \(\sigma\)-biderivations and \(\sigma\)-commuting maps of triangular algebras. Mediterr. J. Math. 14, 2 (2017), 25. Id/No 68
Munkres, J.R.: Elements of Algebraic Topology. Addison-Wesley Publishing Company Inc, California (1984)
Rota, G.-C.: On the foundations of combinatorial theory. I. Theory of Möbius functions. Z. Wahrscheinlichkeitstheorie und Verw. Gebiete 2, 4 (1964), 340–368
Sánchez-Ortega, J.: \(\sigma\)-mappings of triangular algebras. arXiv:1312.4635 (2013)
Spiegel, E., O’Donnell, C.J.: Incidence Algebras. Marcel Dekker, New York (1997)
Stanley, R.: Structure of incidence algebras and their automorphism groups. Bull. Am. Math. Soc. 76, 1236–1239 (1970)
Xiao, Z.K.: Jordan derivations of incidence algebras. Rocky Mountain J. Math. 45(4), 1357–1368 (2015)
Zhang, X., Khrypchenko, M.: Lie derivations of incidence algebras. Linear Algebra Appl. 513, 69–83 (2017)
Acknowledgements
The second author was partially supported by CMUP, member of LASI, which is financed by national funds through FCT—Fundação para a Ciência e a Tecnologia, I.P., under the project with reference UIDB/00144/2020. The authors are grateful to the referee for a very careful reading of the paper and the suggested useful improvements.
Author information
Authors and Affiliations
Corresponding author
Ethics declarations
Conflict of interest
There is no competing interest.
Additional information
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.
About this article
Cite this article
Fornaroli, É.Z., Khrypchenko, M. Skew derivations of incidence algebras. Collect. Math. (2023). https://doi.org/10.1007/s13348-023-00423-7
Received:
Accepted:
Published:
DOI: https://doi.org/10.1007/s13348-023-00423-7