Jump to ContentJump to Main Navigation
Show Summary Details
More options …

Open Mathematics

formerly Central European Journal of Mathematics

Editor-in-Chief: Gianazza, Ugo / Vespri, Vincenzo

1 Issue per year


IMPACT FACTOR 2017: 0.831
5-year IMPACT FACTOR: 0.836

CiteScore 2017: 0.68

SCImago Journal Rank (SJR) 2017: 0.450
Source Normalized Impact per Paper (SNIP) 2017: 0.829

Mathematical Citation Quotient (MCQ) 2016: 0.23

Open Access
Online
ISSN
2391-5455
See all formats and pricing
More options …
Volume 13, Issue 1

Issues

Volume 13 (2015)

Hom-structures on semi-simple Lie algebras

Wenjuan Xie / Quanqin Jin / Wende Liu
Published Online: 2015-10-19 | DOI: https://doi.org/10.1515/math-2015-0059

Abstract

A Hom-structure on a Lie algebra (g,[,]) is a linear map σ W g σ g which satisfies the Hom-Jacobi identity: [σ(x), [y,z]] + [σ(y), [z,x]] + [σ(z),[x,y]] = 0 for all x; y; z ∈ g. A Hom-structure is referred to as multiplicative if it is also a Lie algebra homomorphism. This paper aims to determine explicitly all the Homstructures on the finite-dimensional semi-simple Lie algebras over an algebraically closed field of characteristic zero. As a Hom-structure on a Lie algebra is not necessarily a Lie algebra homomorphism, the method developed for multiplicative Hom-structures by Jin and Li in [J. Algebra 319 (2008): 1398–1408] does not work again in our case. The critical technique used in this paper, which is completely different from that in [J. Algebra 319 (2008): 1398– 1408], is that we characterize the Hom-structures on a semi-simple Lie algebra g by introducing certain reduction methods and using the software GAP. The results not only improve the earlier ones in [J. Algebra 319 (2008): 1398– 1408], but also correct an error in the conclusion for the 3-dimensional simple Lie algebra sl2. In particular, we find an interesting fact that all the Hom-structures on sl2 constitute a 6-dimensional Jordan algebra in the usual way.

Keywords: Hom-structure; Simple Lie algebra; Jordan algebra

References

  • [1] Hartwig J., Larsson D., Silvestrov S., Deformations of Lie algebras using σ-derivations, J. Algebra, 2006, 295, 314-361 Google Scholar

  • [2] Larsson D., Silvestrov S., Quasi-Hom-Lie algebras, central extensions and 2-cocycle-like identities, J. Algebra, 2005, 288, 321-344 Google Scholar

  • [3] Larsson D., Silvestrov S., Graded quasi-Lie algebras, Czechoslovak J. Phys., 2005, 55, 1473-1478 Google Scholar

  • [4] Larsson D., Silvestrov S., Quasi-deformations of sl2.F/ using twisted derivations, Comm. Algebra, 2007, 35, 4303-4318 Google Scholar

  • [5] Jin Q., Li X., Hom-structures on semi-simple Lie algebras, J. Algebra 2008, 319, 1398-1408 Google Scholar

  • [6] Makhlouf A., Silvestrov S., Hom-algebra structures, J. Gen. Lie Theory Appl. 2008, 2 (2), 51-64 Google Scholar

  • [7] Chen Y., Wang Y., Zhang L., The construction of Hom-Lie bialgebras, J. Lie Theory, 2010, 20, 767-783 Google Scholar

  • [8] Sheng Y., Representations of Hom-Lie algebras, Algebr. Represent. Theory, 2012, 15, 1081-1098 Google Scholar

  • [9] Sheng Y., Chen D., Hom-Lie 2-algebras, J. Algebra, 2013, 376, 174-195 Google Scholar

  • [10] Benayadi S., Makhlouf A., Hom-Lie algebras with symmetric invariant nondegenerate bilinear forms, J. Geom. Phys., 2014, 76, 38-60 Web of ScienceGoogle Scholar

  • [11] Hu N., q-Witt algebras, q-Lie algebras, q-holomorph structure and representations, Algebra Colloq., 1999, 6(1), 51-70 Google Scholar

  • [12] Yuan J., Sun L., Liu W., Hom-Lie superalgebra structures on infinite-dimensional simple Lie superalgebras of vector fields, J. Geom. Phys., 2014, 84, 1-7 Web of ScienceGoogle Scholar

  • [13] GAP-groups, algorithms, programming-a system for computational discrete algebra, version 4.7.5, 2014, (http://www.gap-system.org) Google Scholar

  • [14] Humphreys J., Introduction to Lie algebras and representation theory, Springer-Verlag,, New York, 1972 Google Scholar

  • [15] Li X., Li Y., Classification of 3-dimensional multiplicative Hom-Lie algebras, J. Xinyang Normal University, 2012, 25(4), 427-430 Google Scholar

About the article

Received: 2015-03-07

Accepted: 2015-09-03

Published Online: 2015-10-19


Citation Information: Open Mathematics, Volume 13, Issue 1, ISSN (Online) 2391-5455, DOI: https://doi.org/10.1515/math-2015-0059.

Export Citation

©2015 Wenjuan Xie et al.. This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 3.0 License. BY-NC-ND 3.0

Citing Articles

Here you can find all Crossref-listed publications in which this article is cited. If you would like to receive automatic email messages as soon as this article is cited in other publications, simply activate the “Citation Alert” on the top of this page.

[1]
Liqiang Cai and Yunhe Sheng
Science China Mathematics, 2018
[2]
Wenjuan Xie and Wende Liu
Journal of Algebra and Its Applications, 2017, Volume 16, Number 08, Page 1750154

Comments (0)

Please log in or register to comment.
Log in