LEFT-HOM-SYMMETRIC AND HOM-POISSON DIALGEBRAS

Abstract

The aim of this paper is to introduce left-Hom-symmetric dial- gebras (which contain left-Hom-symmetric algebras or Hom-preLie algebras and Hom-dialgebras as special cases) and Hom-Poisson dialgebras. We give some examples and some construction theorems by using the composition con- struction. We prove that the commutator bracket of any left-Hom-symmetric dialgebra provides Hom-Leibniz algebra. We also prove that bimodules over Hom-dialgebras are closed under twisting. Next, we show that bimodules over Hom-dendriform algebras D extend to bimodules over the left-Hom-symmetric algebra associated to D. Finally, we give some examples of Hom-Poisson dial- gebras and prove that the commutator bracket of any Hom-dialgebra structure map leads to Hom-Poisson dialgebra.

Keywords

• Hom-Leibniz algebras,left-Hom-symmetric dialgebras,left-Hom- symmetric algebras,Hom-dendriform algebras

References

1. [1] A. Makhlouf and S. Silvestrov, Hom-algebra structures, J. Gen. Lie Theory Appl. Vol.2 (2008), no. 2, 51-64.
2. [2] A. Makhlouf and D. Yau , Rota-Baxter Hom-Lie admissible algebras, Communication in Algebra, 23, no 3, 1231-1257, 2014.
3. [3] D. Yau, Non-commutative Hom-Poisson algebras, ArXiv : 1010.3408v1, 17 Oct 2010.
4. [4] D. Yau , Envelopping algebras of Hom-Lie algebras , J. Gen. Lie Theory Appl 2(2), 95-108, 2008.
5. [5] D. Yau, Module Hom-algebras, ArXiv:0812.4695v1, 26 Dec 2008.
6. [6] J. Hartwig, D. Larsson and S. Silvestrov, Deformations of Lie algebras using -derivations, J. Algebra 295 (2006), 314-361.
7. [7] J-L Loday, Dialgebras, arXiv : math/0102053v1, 7 Feb 2001.
8. [8] J. L. Loday, Une version non commutative des algebres de Lie: les algebres de Leibniz, Ens. Math., 39 (1993), 269-293.

9. [9] M. Aguiar, Innitesimal bialgebras, pre-Lie and dendriform algebras, arXiv:math/0211074v3 [Math.QA] 16 Nov 2002.
10. [10] R. Felipe, A brief fondation of the left symmetric dialgebras, Comminicacion del CIMAT No I-11-02/18-03-2011 (MB/CIMAT).
11. [11] S. Benayadi and S. Hidri Quadratic Leibniz Algebras, Journal of Lie Theory, 24 (2014) 737-759.