In
mathematical physics
Mathematical physics refers to the development of mathematics, mathematical methods for application to problems in physics. The ''Journal of Mathematical Physics'' defines the field as "the application of mathematics to problems in physics and t ...
, the ternary commutator is an additional
ternary operation
In mathematics, a ternary operation is an ''n''- ary operation with ''n'' = 3. A ternary operation on a set ''A'' takes any given three elements of ''A'' and combines them to form a single element of ''A''.
In computer science, a ternary operator ...
on a
triple system
In algebra, a triple system (or ternar) is a vector space ''V'' over a field F together with a F-trilinear map
: (\cdot,\cdot,\cdot) \colon V\times V \times V\to V.
The most important examples are Lie triple systems and Jordan triple systems. The ...
defined by
:
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
In mathematics, the commutator gives an indication of the extent to which a certain binary operation fails to be commutative. There are different definitions used in group theory and ring theory.
Group theory
The commutator of two elements, a ...
.
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
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Algebras
Ternary operations
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