In mathematical physics, the ternary commutator is an additional ternary operation on a triple system defined by

${\displaystyle [a,b,c]=abc-acb-bac+bca+cab-cba.\,}$

Also called the ternutator or alternating ternary sum, it is a special case of the n-commutator for n = 3, whereas the 2-commutator is the ordinary commutator.

Properties

• When one or more of a, b, c is equal to 0, [a, b, c] is also 0. This statement makes 0 the absorbing element of the ternary commutator.
  • The same happens when a = b = c.

Further reading

• Bremner, Murray R. (15 August 1998), "Identities for the Ternary Commutator", Journal of Algebra, 206 (2): 615–623, doi:10.1006/jabr.1998.7433
• Bremner, Murray R.; Ortega, Juana Sánchez (25 October 2010), "The partially alternating ternary sum in an associative dialgebra", Journal of Physics A: Mathematical and Theoretical, 43 (56): 455215, arXiv:1008.2721, Bibcode:2010JPhA...43S5215B, doi:10.1088/1751-8113/43/45/455215, S2CID 6636902
• Bremner, Murray R.; Peresi, Luiz A. (1 April 2006), "Ternary analogues of Lie and Malcev algebras", Linear Algebra and Its Applications, 414 (1): 1–18, doi:10.1016/j.laa.2005.09.004
• Bremner, Murray R.; Peresi, Luiz A. (26 July 2012), "Higher identities for the ternary commutator", Journal of Physics A: Mathematical and General, 45 (50): 505201, arXiv:1207.6312, Bibcode:2012JPhA...45X5201B, doi:10.1088/1751-8113/45/50/505201, S2CID 17037773
• Devchand, Chandrashekar; Fairlie, David; Nuyts, Jean; Weingart, Gregor (6 November 2009), "Ternutator identities", Journal of Physics A: Mathematical and Theoretical, 42 (47): 475209, arXiv:0908.1738, Bibcode:2009JPhA...42U5209D, doi:10.1088/1751-8113/42/47/475209, S2CID 17246666
• Nambu, Yoichiro (1973), "Generalized Hamiltonian Dynamics", Physical Review D, 7 (8): 2405–2412, Bibcode:1973PhRvD...7.2405N, doi:10.1103/PhysRevD.7.2405