Login
Login

Ternary commutator

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

:[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

  • {{Citation |last=Bremner |first=Murray R. |date=15 August 1998 |title=Identities for the Ternary Commutator |journal=Journal of Algebra |volume=206 |issue=2 |pages=615–623 |doi=10.1006/jabr.1998.7433 |doi-access=free }}
  • {{Citation |last1=Bremner |first1=Murray R. |last2=Ortega |first2=Juana Sánchez |date=25 October 2010 |title=The partially alternating ternary sum in an associative dialgebra |journal=Journal of Physics A: Mathematical and Theoretical |volume=43 |issue=56 |page=455215 |doi=10.1088/1751-8113/43/45/455215 |arxiv=1008.2721|bibcode=2010JPhA...43S5215B |s2cid=6636902 }}
  • {{Citation |last1=Bremner |first1=Murray R. |last2=Peresi |first2=Luiz A. |date=1 April 2006 |title=Ternary analogues of Lie and Malcev algebras |journal=Linear Algebra and Its Applications |volume=414 |issue=1 |pages=1–18 |doi=10.1016/j.laa.2005.09.004 |doi-access=free }}
  • {{Citation |last1=Bremner |first1=Murray R. |last2=Peresi |first2=Luiz A. |date=26 July 2012 |title=Higher identities for the ternary commutator |journal=Journal of Physics A: Mathematical and General |volume=45 |issue=50 |page=505201 |doi=10.1088/1751-8113/45/50/505201 |arxiv=1207.6312|bibcode=2012JPhA...45X5201B |s2cid=17037773 }}
  • {{Citation |last1=Devchand |first1=Chandrashekar |last2=Fairlie |first2=David |last3=Nuyts |first3=Jean |last4=Weingart |first4=Gregor |date=6 November 2009 |title=Ternutator identities |journal=Journal of Physics A: Mathematical and Theoretical |volume=42 |issue=47 |page=475209 |doi=10.1088/1751-8113/42/47/475209 |arxiv=0908.1738|bibcode=2009JPhA...42U5209D |s2cid=17246666 }}
  • {{Citation |last=Nambu |first=Yoichiro |authorlink=Yoichiro Nambu |year=1973 |title=Generalized Hamiltonian Dynamics |journal=Physical Review D |volume=7 |issue=8 |pages=2405–2412 |doi=10.1103/PhysRevD.7.2405|bibcode=1973PhRvD...7.2405N }}

Category:Algebras

Category:Ternary operations

{{abstract-algebra-stub}}