Hom-structures on semi-simple Lie algebras

• Autorzy: Wenjuan Xie , Quanqin Jin , Wende Liu
• Języki publikacji: EN

Abstrakty

EN

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.

Słowa kluczowe

EN

Hom-structure; Simple Lie algebra; Jordan algebra

• Wydawca: De Gruyter Open
• Czasopismo: Open Mathematics
• Rocznik: 2015
• Tom: 13
• Numer: 1

Daty

otrzymano

2015-03-07

zaakceptowano

2015-09-03

online

2015-10-19

Twórcy

autor

Wenjuan Xie

• School of Mathematical Sciences, Harbin Normal University, Harbin 150025, China

autor

Quanqin Jin

• Department of Mathematics, Tongji University, Shanghai 200092, China

autor

Wende Liu

• School of Mathematical Sciences, Harbin Normal University, Harbin 150025, China

Bibliografia

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Identyfikatory

DOI

10.1515/math-2015-0059