Main Article Content
The aim of this work is to introduce and study the notions of Hom-pre-Jordan algebra and Hom-J-dendriform algebra which generalize Hom-Jordan algebras. Hom-pre-Jordan algebras are regarded as the underlying algebraic structures of the Hom-Jordan algebras behind the Rota-Baxter operators and O-operators introduced in this paper. Hom-pre-Jordan algebras are also analogues of Hom-pre-Lie algebras for Hom-Jordan algebras. The anti-commutator of a Hom-pre-Jordan algebra is a Hom-Jordan algebra and the left multiplication operator gives a representation of a Hom-Jordan algebra. On the other hand, a Hom-J-dendriform algebra is a Hom-Jordan algebraic analogue of a Hom-dendriform algebra such that the anti-commutator of the sum of the two operations is a Hom-pre-Jordan algebra.
C. Bai, O. Bellier, L. Guo, X. Ni, Splitting of operations, Manin products, and Rota-Baxter operators, Int. Math. Res. Not. IMRN 2013 (3) (2013), 485 – 524.
C.-H. Chu, Jordan triples and Riemannian symmetric spaces, Adv. Math. 219 (2008), 2029 – 2057.
J.T. Hartwig, D. Larsson, S.D. Silvestrov, Deformations of Lie algebras using σ-derivations, J. Algebra 295 (2006), 314 – 361.
D. Hou, C. Bai, J-dendriform algebras, Front. Math. China 7 (2012) (1), 29 – 49.
D. Hou, X. Ni, C. Bai, Pre-Jordan algebras, Math. Scand. 112 (1) (2013), 19 – 48.
N. Jacobson, Lie and Jordan triple systems, Amer. J. Math. 71 (1949), 149 – 170.
J.-L. Loday, M. Ronco, Trialgebras and families of polytopes, in “ Homotopy Theory: relations with Algebraic Geometry, Group Cohomology, and Algebraic K-Theory ”, Contemp. Math. 346, Amer. Math. Soc., Providence, RI, 2004, 369 – 673.
J.-L. Loday, “ Dialgebras ”, Lecture Notes in Math. 1763, Springer, Berlin, 2001, 7 – 66.
A. Makhlouf, Hom-dendriform algebras and Rota-Baxter Hom-algebras, in “ Operads And Universal Algebra ”, Nankai Ser. Pure Appl. Math. Theoret. Phys. 9, World Sci. Publ., Hackensack, NJ, 2012, 147 – 171.
A. Makhlouf, Hom-alternative algebras and Hom-Jordan algebras, Int. Electron. J. Algebra 8 (2010), 177 – 190.
Q. Sun, On Hom-prealternative bialgebras, Algebr. Represent. Theory 19 (3) (2016), 657 – 677.
Y. Sun, Z. Huang, S. Zhao, Z. Tian, Classification of pre-Jordan Algebras and Rota-Baxter Operators on Jordan Algebras in Low Dimensions, 2021, arXiv:2111.02035.
H. Upmeier, Jordan algebras and harmonic analysis on symmetric spaces, Amer. J. Math. 108 (1986), 1 – 25.
D. Yau, Hom-Maltsev, Hom-alternative and Hom-Jordan algebras, Int. Electron. J. Algebra 11 (2012), 177 – 217.