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.
📌 3 Key Takeaways
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Objective & Hypothesis: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.
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Key Findings & Data: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.
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Conclusion & Impact: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.
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.
F. V. Atkinson, Some aspects of Baxters functional equation, J. Math. Anal. Appl. 7, 1-30, 1963.
C.M. Bai, A unified algebraic approach to classical Yang-Baxter equation, J. Phy. A: Math. Theor., 40, 11073-11082, 2007.
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.
C. Bai and D.P. Hou, J-dendriform algebras, Front. Math. China. 7 (1), 29-49, 2012.
C. Bai, L.G. Liu and X. Ni, Some results on L-dendriform algebras, J. Geom. Phys. 60 (6-8), 940-950, 2010.
C. Bai and X. Ni, Pre-alternative algebras and pre-alternative bialgebras, Pacific J. Math. 248, 355-390, 2010.
G. Baxter, An analytic problem whose solution follows from a simple algebraic identity, Pacific J. Math. 10, 731-742, 1960.
D. Burde and K. Dekimpe, Post-Lie algebra structures on pairs of Lie algebras, J. Algebra, 464, 226-245, 2016.
P. Cartier, On the structure of free Baxter algebras, Adv. Math. 9, 253-265, 1972.
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.
M.E. Goncharov, Structures of Malcev bialgebras on a simple non-Lie Malcev Algebra, Commun. Algebra 40 (8), 3071-3094, 2012.
V. Yu. Gubarev and P.S. Kolesnikov, Operads of decorated trees and their duals, Comment. Math. Univ. Carolin. 55 (4), 421-445, 2014.
L. Guo, What is a RotaBaxter algebra, Notices. Amer. Math. Soc. 56, 14361437, 2009.
L. Guo and W. Keigher, Baxter algebras and shuffle products, Adv. Math. 150, 117149, 2000.
L. Guo and B. Zhang, Renormalization of multiple zeta values, J. Algebra 319, 37703809, 2008.
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
D. Hou, X. Ni and C. Bai, Pre-Jordan algebras, Math. Scand. 112 (1), 19-48, 2013.
F.S. Kerdman, Analytic Moufang loops in the large, Algebra Log. 18, 325-347, 1980.
B.A. Kupershmidt, What a Classical r-Matrix Really Is, J. Nonlin. Math. Phys. 6 (4), 448-488, 1999.
E.N. Kuzmin, Malcev algebras and their representations, Algebra Log. 7 233-244, 1968.
E.N. Kuzmin, The connection between Malcev algebras and analytic Moufang loops, Algebra Log. 10, 1-14, 1971.
L. Liu, X. Ni and C. Bai, L-quadri-algebras, Scientia Sinica Mathematica, 41 (2), 105-124, 2011.
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.
J.-L. Loday and M. Ronco, Trialgebras and families of polytopes. Contemp. Math. 346, 369-398, 2004.
S. Madariaga,Splitting of operations for alternative and Malcev structures, Commun. Algebra, 45 (1), 183-197, 2014.
A.I. Malcev, Analytic loops, Mat. Sb. 36, 569-576, 1955.
P.T. Nagy, Moufang loops and Malcev algebras, Sem. Sophus Lie 3, 65-68, 1993.
P.C. Rosenbloom, Post Algebras. I. Postulates and General Theory, Amer. J. Math. 64 (1), 167-188, 1942.
G.-C. Rota, Baxter algebras and combinatorial identities I, Bull. Amer. Math. Soc. 75, 325-329, 1969.
G. Rousseau, Post algebras and pseudo-Post algebras, Fundamenta Mathematicae, 67 133-145, 1970.
R. D. Schafer, Representations of alternative algebras, Trans. Amer. Math. Soc. 72, 1-17, 1952.
B. Vallette, Homology of generalized partition posets, J. Pure Appl. Algebra, 208 (2), 699-725, 2007.
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.
G. Baxter, An analytic problem whose solution follows from a simple algebraic identity, Pacific J. Math. 10, 731-742, 1960.
V. Yu. Gubarev and P.S. Kolesnikov, Operads of decorated trees and their duals, Comment. Math. Univ. Carolin. 55 (4), 421-445, 2014.
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.