×

Rota-Baxter multiplicative 3-ary Hom-Nambu-Lie algebras. (English) Zbl 1368.17004

Summary: In this paper, we introduce the concepts of Rota-Baxter operators and differential operators with weights on a multiplicative \(n\)-ary Hom-algebra. We then focus on Rota-Baxter multiplicative 3-ary Hom-Nambu-Lie algebras and show that they can be derived from Rota-Baxter Hom-Lie algebras, Hom-preLie algebras and Rota-Baxter commutative Hom-associative algebras. We also explore the connections between these Rota-Baxter multiplicative 3-ary Hom-Nambu-Lie algebras.

MSC:

17A42 Other \(n\)-ary compositions \((n \ge 3)\)
17B99 Lie algebras and Lie superalgebras
PDFBibTeX XMLCite
Full Text: DOI arXiv

References:

[1] Arnlind, J.; Makhlouf, A.; Silvestrov, S., Construction of \(n\)-Lie algebras and \(n\)-ary Hom-Nambu-Lie algebras, J. Math. Phys., 52 (2011), 123502, 13 pp · Zbl 1273.17005
[2] Baxter, G., An analytic problem whose solution follows from a simple algebraic identity, Pacific J. Math., 10, 731-742 (1960) · Zbl 0095.12705
[3] Cartier, P., On the structure of free Baxter algebras, Adv. Math., 9, 253-265 (1972) · Zbl 0267.60052
[4] Rota, G.-C., Baxter algebras and combinatorial identities I, II, Bull. Amer. Math. Soc., 75, 325-334 (1969) · Zbl 0192.33801
[5] Rota, G.-C., Baxter operators, an introduction. Gian-Carlo Rota on combinatorics, (Contemp. Mathematicians (1995), Birkhäuser Boston: Birkhäuser Boston Boston, MA), 504-512
[6] Aguiar, M., Pre-Poisson algebras, Lett. Math. Phys., 54, 263-277 (2000) · Zbl 1032.17038
[7] Bai, C.; Guo, L.; Ni, X., Generalizations of the classical Yang-Baxter equation and O-operators, J. Math. Phys., 52, Article 063515 pp. (2011) · Zbl 1317.16031
[8] Guo, L., What is a Rota-Baxter algebra, Notices Amer. Math. Soc., 56, 1436-1437 (2009) · Zbl 1184.16038
[9] Guo, L., Introduction to Rota-Baxter Algebra (2012), International Press and Higher Education Press · Zbl 1271.16001
[10] Guo, L.; Keigher, W., Baxter algebras and shuffle products, Adv. Math., 150, 117-149 (2000) · Zbl 0947.16013
[11] Guo, L.; Sit, W.; Zhang, R., Differemtail type operators and Gröbner-Shirshov bases, J. Symolic Comput., 52, 97-123 (2013) · Zbl 1290.16021
[12] Guo, L.; Zhang, B., Renormalization of multiple zeta values, J. Algebra, 319, 3770-3809 (2008) · Zbl 1165.11071
[13] Connes, A.; Kreimer, D., Hopf algebras, renormalisation and noncommutative geometry, Comm. Math. Phys., 199, 203-242 (1988) · Zbl 0932.16038
[14] Ebrahimi-Fard, K.; Guo, L.; Kreimer, D., Spitzer’s identity and the algebraic Birkhoff decomposition in pQFT, J. Phys. A, 37, 11037-11052 (2004) · Zbl 1062.81113
[15] Ebrahimi-Fard, K.; Guo, L.; Manchon, D., Birkhoff type decompositions and the Baker-Campbell-Hausdorff recursion, Comm. Math. Phys., 267, 821-845 (2006) · Zbl 1188.17020
[16] Manchon, D.; Paycha, S., Nested sums of symbols and renormalised multiple zeta values, Int. Math. Res. Not., 4628C4697 (2010)
[17] Bai, C.; Guo, L.; Ni, X., Nonabelian generalized Lax pairs, the classical Yang-Baxter equation and PostLie algebras, Comm. Math. Phys., 297, 553-596 (2010) · Zbl 1206.17020
[18] Semenov-Tyan-Shanskii, M., What a classical \(r\)-matrix is, Funktsional. Anal. i Prilozhen., 17, 17-33 (1983)
[19] Bai, R.; Guo, L.; Li, J.; Wu, Y., Rota-Baxter 3-Lie algebras, J. Math. Phys., 54, Article 063504 pp. (2013) · Zbl 1366.17003
[20] An, H.; Bai, C., From Rota-Baxter algebras to pre-Lie algebras, J. Phys. A, 41 (2008), 015201, 19 pp · Zbl 1132.81031
[21] Bai, R.; Bai, C.; Wang, J., Realizations of 3-Lie algebras, J. Math. Phys., 51, Article 063505 pp. (2010) · Zbl 1311.17002
[22] Li, X.; Hou, D.; Bai, C., Rota-Baxter operators on pre-Lie algebras, J. Nonlinear Math. Phys., 14, 269-289 (2007) · Zbl 1203.17016
[23] Makhlouf, A.; Donald, Y., Rota-Baxter Hom-Lie-admissible algebras, Comm. Algebra., 42, 1231-1257 (2014) · Zbl 1367.17003
[24] Yau, D., On \(n\)-ary Hom-Nambu and Hom-Nambu-Lie algebras, J. Geom. Phys., 62, 506-522 (2012) · Zbl 1280.17006
[25] Makhlouf, A., Hom-dendriform algebras and Rota-Baxter Hom-algebras, Nankai Ser. Pure Appl. Math. Theoret. Phys., 9, 147-171 (2012) · Zbl 1351.17035
[26] Arnlind, J.; Makhlouf, A.; Silvestrov, S., Ternary Hom-Nambu-Lie algebras induced by Hom-Lie algebras, J. Math. Phys., 51 (2010), 043515, 11 pp · Zbl 1310.17001
[28] Lister, W., A structure theory of Lie triple systems, Trans. Amer. Math. Soc., 72, 217-242 (1952) · Zbl 0046.03404
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.