Explicit 4-term commutator $[A,B,C,D]$

Question

The commutator of two terms is:

$$ [A,B]=AB-BA $$

The commutator of three terms is:

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

I was not able to google the four term commutator, not am I able to construct it from the general definition. For reference, can any one explicitly state the expression of the four term commutator?

Answer 1

Based on the formulas you've presented, I suspect that the general formula is $$ [A_1,A_2,\dots,A_n] = \sum_{\sigma \in S_n} \operatorname{sgn}(\sigma)\prod_{i=1}^n A_{\sigma_i}, $$ which is a formula analogous to the Leibniz formula for the determinant. Thus, your four-fold commutator would be a sum of 24 terms, 12 with a $+$ and 12 with a $-$.

The terms with a $+$ will have indices in the order $$ 1234,3124,2314,4132,2431,4213,3241,1423,1342,2143,3412,4321 $$ (corresponding to all elements of the alternating group $A_4$) and the remaining orders of indices, namely $$ 2134,1324,3214,1432,4231,2413,2341,4123,3142,1243,4312,3421 $$ will have a $-$.

Answer 2

I'm not familiar with the term "$k$-term commutator", but a sensible definition that specializes for $k = 2, 3$ to the given pattern is $$\boxed{[A_1, \ldots, A_k] = \sum_{\sigma \in S_k} \operatorname{sgn}(\sigma) A_{\sigma(1)} \cdots A_{\sigma(k)}} .$$ Here, $S_n$ denotes the group of permutations of $\{1, \ldots, k\}$.

Expanding this expression gives $k!$ terms---already $24$ terms in the $k = 4$ case, and so I won't write out all of them---but sorting terms by lexicographical order gives the expansion $$[A, B, C, D] = ABCD - ABDC - ACBD + ACDB + ADBC - ADCB + \cdots .$$ Note that the explicitly written terms comprise $A[B, C, D]$, and indeed we have $$[A, B, C, D] = A[B, C, D] - B[C, D, A] + C[D, A, B] - D[A, B, C] .$$ We can generalize this observation: A standard induction argument shows that we could inductively define the "$k$-term commutator" by declaring (if the underlying algebra has an identity $1$) $[] = 1$ or (in general) $[A] = A$, and $$[A_1, \ldots, A_k] = \mathfrak{S} (-1)^{(i - 1)(k - 1)} A_1 [A_2, \ldots, A_k]$$ where $\mathfrak{S}$ denotes a sum over the cyclic permutations of $(1, \ldots, k)$, and where $i$ denotes the image of $1$ under a given permutation.