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Hom-structures on semi-simple Lie algebras

Treść / Zawartość
Warianty tytułu
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
Wydawca
Czasopismo
Rocznik
Tom
13
Numer
1
Opis fizyczny
Daty
otrzymano
2015-03-07
zaakceptowano
2015-09-03
online
2015-10-19
Twórcy
autor
  • School of Mathematical Sciences, Harbin Normal University, Harbin 150025, China
autor
  • Department of Mathematics, Tongji University, Shanghai 200092, China
autor
  • School of Mathematical Sciences, Harbin Normal University, Harbin 150025, China
Bibliografia
  • [1] Hartwig J., Larsson D., Silvestrov S., Deformations of Lie algebras using σ-derivations, J. Algebra, 2006, 295, 314-361
  • [2] Larsson D., Silvestrov S., Quasi-Hom-Lie algebras, central extensions and 2-cocycle-like identities, J. Algebra, 2005, 288,321-344
  • [3] Larsson D., Silvestrov S., Graded quasi-Lie algebras, Czechoslovak J. Phys., 2005, 55, 1473-1478
  • [4] Larsson D., Silvestrov S., Quasi-deformations of sl2.F/ using twisted derivations, Comm. Algebra, 2007, 35, 4303-4318
  • [5] Jin Q., Li X., Hom-structures on semi-simple Lie algebras, J. Algebra 2008, 319, 1398-1408
  • [6] Makhlouf A., Silvestrov S., Hom-algebra structures, J. Gen. Lie Theory Appl. 2008, 2 (2), 51-64
  • [7] Chen Y., Wang Y., Zhang L., The construction of Hom-Lie bialgebras, J. Lie Theory, 2010, 20, 767-783
  • [8] Sheng Y., Representations of Hom-Lie algebras, Algebr. Represent. Theory, 2012, 15, 1081-1098
  • [9] Sheng Y., Chen D., Hom-Lie 2-algebras, J. Algebra, 2013, 376, 174-195
  • [10] Benayadi S., Makhlouf A., Hom-Lie algebras with symmetric invariant nondegenerate bilinear forms, J. Geom. Phys., 2014, 76,38-60[WoS]
  • [11] Hu N., q-Witt algebras, q-Lie algebras, q-holomorph structure and representations, Algebra Colloq., 1999, 6(1), 51-70
  • [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]
  • [13] GAP-groups, algorithms, programming-a system for computational discrete algebra, version 4.7.5, 2014,(http://www.gap-system.org)
  • [14] Humphreys J., Introduction to Lie algebras and representation theory, Springer-Verlag,, New York, 1972
  • [15] Li X., Li Y., Classification of 3-dimensional multiplicative Hom-Lie algebras, J. Xinyang Normal University, 2012, 25(4), 427-430
Typ dokumentu
Bibliografia
Identyfikatory
Identyfikator YADDA
bwmeta1.element.doi-10_1515_math-2015-0059
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