Hom-structures on semi-simple Lie algebras

Wenjuan Xie; Quanqin Jin; Wende Liu

Open Mathematics (2015)

  • Volume: 13, Issue: 1
  • ISSN: 2391-5455

Abstract

top
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.

How to cite

top

Wenjuan Xie, Quanqin Jin, and Wende Liu. "Hom-structures on semi-simple Lie algebras." Open Mathematics 13.1 (2015): null. <http://eudml.org/doc/275848>.

@article{WenjuanXie2015,
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.},
author = {Wenjuan Xie, Quanqin Jin, Wende Liu},
journal = {Open Mathematics},
keywords = {Hom-structure; Simple Lie algebra; Jordan algebra},
language = {eng},
number = {1},
pages = {null},
title = {Hom-structures on semi-simple Lie algebras},
url = {http://eudml.org/doc/275848},
volume = {13},
year = {2015},
}

TY - JOUR
AU - Wenjuan Xie
AU - Quanqin Jin
AU - Wende Liu
TI - Hom-structures on semi-simple Lie algebras
JO - Open Mathematics
PY - 2015
VL - 13
IS - 1
SP - null
AB - 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.
LA - eng
KW - Hom-structure; Simple Lie algebra; Jordan algebra
UR - http://eudml.org/doc/275848
ER -

References

top
  1. [1] Hartwig J., Larsson D., Silvestrov S., Deformations of Lie algebras using σ-derivations, J. Algebra, 2006, 295, 314-361 Zbl1138.17012
  2. [2] Larsson D., Silvestrov S., Quasi-Hom-Lie algebras, central extensions and 2-cocycle-like identities, J. Algebra, 2005, 288, 321-344 Zbl1099.17015
  3. [3] Larsson D., Silvestrov S., Graded quasi-Lie algebras, Czechoslovak J. Phys., 2005, 55, 1473-1478 
  4. [4] Larsson D., Silvestrov S., Quasi-deformations of sl2.F/ using twisted derivations, Comm. Algebra, 2007, 35, 4303-4318 Zbl1131.17010
  5. [5] Jin Q., Li X., Hom-structures on semi-simple Lie algebras, J. Algebra 2008, 319, 1398-1408 Zbl1144.17005
  6. [6] Makhlouf A., Silvestrov S., Hom-algebra structures, J. Gen. Lie Theory Appl. 2008, 2 (2), 51-64 
  7. [7] Chen Y., Wang Y., Zhang L., The construction of Hom-Lie bialgebras, J. Lie Theory, 2010, 20, 767-783 Zbl1217.17013
  8. [8] Sheng Y., Representations of Hom-Lie algebras, Algebr. Represent. Theory, 2012, 15, 1081-1098 Zbl1294.17001
  9. [9] Sheng Y., Chen D., Hom-Lie 2-algebras, J. Algebra, 2013, 376, 174-195 Zbl1281.17034
  10. [10] Benayadi S., Makhlouf A., Hom-Lie algebras with symmetric invariant nondegenerate bilinear forms, J. Geom. Phys., 2014, 76, 38-60 [WoS] Zbl1331.17028
  11. [11] Hu N., q-Witt algebras, q-Lie algebras, q-holomorph structure and representations, Algebra Colloq., 1999, 6(1), 51-70 Zbl0943.17007
  12. [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 [WoS] Zbl06319463
  13. [13] GAP-groups, algorithms, programming-a system for computational discrete algebra, version 4.7.5, 2014, (http://www.gap-system.org) 
  14. [14] Humphreys J., Introduction to Lie algebras and representation theory, Springer-Verlag,, New York, 1972 Zbl0254.17004
  15. [15] Li X., Li Y., Classification of 3-dimensional multiplicative Hom-Lie algebras, J. Xinyang Normal University, 2012, 25(4), 427-430 Zbl1274.17038

NotesEmbed ?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

    
                

                
Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.