Fattoum HARRATHİ, Sami MABROUK, Othmen NCİB, Sergei SILVESTROV

Malcev Yang-Baxter equation, weighted $\mathcal{O}$-operators on Malcev algebras and post-Malcev algebras

The purpose of this paper is to study the $\mathcal{O}$-operators on Malcev algebras and discuss the solutions of Malcev Yang-Baxter equation by $\mathcal{O}$-operators. Furthermore we introduce the notion of weighted $\mathcal{O}$-operators on Malcev algebras, which can be characterized by graphs of the semi-direct product Malcev algebra. Then we introduce a new algebraic structure called post-Malcev algebras. Therefore, post-Malcev algebras can be viewed as the underlying algebraic structures of weighted $\mathcal{O}$-operators on Malcev algebras. A post-Malcev algebra also gives rise to a new Malcev algebra. Post-Malcev algebras are analogues for Malcev algebras of post-Lie algebras and fit into a bigger framework with a close relationship with post-alternative algebras.

Keywords:

Alternative algebra, Malcev algebra weighted $\mathcal{O}$-operators, Rota-Baxter operator, representation, Malcev Yang-Baxter equation.,

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[1] F. V. Atkinson, Some aspects of Baxters functional equation, J. Math. Anal. Appl. 7, 1-30, 1963.
[2] C.M. Bai, A unified algebraic approach to classical Yang-Baxter equation, J. Phy. A: Math. Theor., 40, 11073-11082, 2007.
[3] C. Bai, O. Bellier, L. Guo and X. Ni, Splitting of operations, Manin products, and Rota-Baxter operators, Int. Math. Res. Notes 3, 485-524, 2013.
[4] C. Bai and D.P. Hou, J-dendriform algebras, Front. Math. China. 7 (1), 29-49, 2012.
[5] C. Bai, L.G. Liu and X. Ni, Some results on L-dendriform algebras, J. Geom. Phys. 60 (6-8), 940-950, 2010.
[6] C. Bai and X. Ni, Pre-alternative algebras and pre-alternative bialgebras, Pacific J. Math. 248, 355-390, 2010.
[7] G. Baxter, An analytic problem whose solution follows from a simple algebraic identity, Pacific J. Math. 10, 731-742 , 1960.
[8] D. Burde and K. Dekimpe, Post-Lie algebra structures on pairs of Lie algebras, J. Algebra, 464, 226-245, 2016.
[9] P. Cartier, On the structure of free Baxter algebras, Adv. Math. 9, 253-265, 1972.
[10] K. Ebrahimi-Fard, A. Lundervold and H. Munthe-Kaas, On the Lie enveloping algebra of a post-Lie algebra, J. Lie Theory 25 (4), 1139-1165, 2015.
[11] M.E. Goncharov, Structures of Malcev bialgebras on a simple non-Lie Malcev Algebra, Commun. Algebra 40 (8), 3071-3094, 2012.
[12] V. Yu. Gubarev and P.S. Kolesnikov, Operads of decorated trees and their duals, Comment. Math. Univ. Carolin. 55 (4), 421-445 , 2014.
[13] L. Guo, What is a RotaBaxter algebra, Notices. Amer. Math. Soc. 56, 14361437, 2009.
[14] L. Guo and W. Keigher, Baxter algebras and shuffle products, Adv. Math. 150, 117149, 2000.
[15] L. Guo and B. Zhang, Renormalization of multiple zeta values, J. Algebra 319, 37703809, 2008.
[16] F. Harrathi, S. Mabrouk, O. Ncib and S. Silvestrov, Kupershmidt operators on Hom- Malcev algebras and their deformation, Int. J. Geom. Methods Mod. Phys. 2022. https://doi.org/10.1142/S0219887823500469
[17] D. Hou, X. Ni and C. Bai, Pre-Jordan algebras, Math. Scand. 112 (1), 19-48, 2013.
[18] F.S. Kerdman, Analytic Moufang loops in the large, Algebra Log. 18, 325-347, 1980.
[19] B.A. Kupershmidt, What a Classical r-Matrix Really Is, J. Nonlin. Math. Phys. 6 (4), 448-488, 1999.
[20] E.N. Kuzmin, Malcev algebras and their representations, Algebra Log. 7 233-244, 1968.
[21] E.N. Kuzmin, The connection between Malcev algebras and analytic Moufang loops, Algebra Log. 10, 1-14, 1971.
[22] E.N. Kuzmin and I.P. Shestakov, Non-associative structures, Algebra VI, Encyclopaedia Math. Sci. 57, Springer, Berlin, 197-280, 1995.
[23] L. Liu, X. Ni and C. Bai, L-quadri-algebras, Scientia Sinica Mathematica, 41 (2), 105-124, 2011.
[24] J.-L. Loday, Dialgebras, in: J.-L. Loday A. Frabetti F. Chapoton F. Goichot (eds.), Dialgebras and Related Operads, Lecture Notes in Mathematics, 1763, 7-66, 2001.
[25] J.-L. Loday and M. Ronco, Trialgebras and families of polytopes. Contemp. Math. 346, 369-398, 2004.
[26] S. Madariaga,Splitting of operations for alternative and Malcev structures, Commun. Algebra, 45 (1), 183-197, 2014.
[27] A.I. Malcev, Analytic loops, Mat. Sb. 36, 569-576, 1955.
[28] P.T. Nagy, Moufang loops and Malcev algebras, Sem. Sophus Lie 3, 65-68, 1993.
[29] P.C. Rosenbloom, Post Algebras. I. Postulates and General Theory, Amer. J. Math. 64 (1), 167-188, 1942.
[30] G.-C. Rota, Baxter algebras and combinatorial identities I, Bull. Amer. Math. Soc. 75, 325-329, 1969.
[31] G. Rousseau, Post algebras and pseudo-Post algebras, Fundamenta Mathematicae, 67 133-145, 1970.
[32] R. D. Schafer, Representations of alternative algebras, Trans. Amer. Math. Soc. 72, 1-17, 1952.
[33] B. Vallette, Homology of generalized partition posets, J. Pure Appl. Algebra, 208 (2), 699-725, 2007.
[34] P. Yu, Q. Liu, C. Bai and L. Guo, Post-Lie algebra structures on the Lie algebra $\mathrm{sl}(2,\mathbb{C})$ , Electron. J. Linear Algebra 23, 180-197, 2012.