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arsmath
arsmath
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Generalizing the commutator and anitanti-commutator

I was wondering if there's any attmeptattempt to generalize the commutator for something general for more than two terms.

Here's what I was thinking of, for $[A,B]=AB-BA$, so for three terms:

$[A,B,C] = ABC+BCA+CAB-BAC-CBA-ACB$

where I have taken a plus sign for a cyclic permutation of $ABC$, and a minus sign for acyclic permutation of $ABC$.

Is such a generlaizationgeneralization still called commutator or something else?

Generalizing the commutator and anit-commutator

I was wondering if there's any attmept to generalize the commutator for something general for more than two terms.

Here's what I was thinking of, for $[A,B]=AB-BA$, so for three terms:

$[A,B,C] = ABC+BCA+CAB-BAC-CBA-ACB$

where I have taken a plus sign for a cyclic permutation of $ABC$, and a minus sign for acyclic permutation of $ABC$.

Is such a generlaization still called commutator or something else?

Generalizing the commutator and anti-commutator

I was wondering if there's any attempt to generalize the commutator for something general for more than two terms.

Here's what I was thinking of, for $[A,B]=AB-BA$, so for three terms:

$[A,B,C] = ABC+BCA+CAB-BAC-CBA-ACB$

where I have taken a plus sign for a cyclic permutation of $ABC$, and a minus sign for acyclic permutation of $ABC$.

Is such a generalization still called commutator or something else?

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Alan
Alan
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Generalizing the commutator and anit-commutator

I was wondering if there's any attmept to generalize the commutator for something general for more than two terms.

Here's what I was thinking of, for $[A,B]=AB-BA$, so for three terms:

$[A,B,C] = ABC+BCA+CAB-BAC-CBA-ACB$

where I have taken a plus sign for a cyclic permutation of $ABC$, and a minus sign for acyclic permutation of $ABC$.

Is such a generlaization still called commutator or something else?