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Suppose we have a function $f(x,y)$ in two variables. Is there an operator on the function, say, $$([]f)(x,y) = f(x,y) - f(y,x)?$$

In other words, I'm looking for a commutator in terms of function arguments.

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By analogy with the odd part of a function and the antisymmetric part of a matrix, you might say that your operator gives twice the anticommutative part of the function. –  Rahul Feb 5 '12 at 11:59
I don't know of any general notation either, but I propose $[x, y]_f$. Note, $[x, y] = xy - yx$ is the ring-theoretic commutator, based on the multiplication function, $m(x,y) = xy$, so the new notation would agree well with established notation: $[x, y]_m = xy - yx = [x, y]$. Just a thought. –  Shaun Ault Feb 5 '12 at 13:55
I really like $[x,y]_f$. –  Peteris Feb 5 '12 at 21:52
@vrich: And what does that have to do with the question? –  Johannes Kloos Mar 2 '13 at 14:27
@vrich: I don't understand this comment: $f(x,y)=e^{ixy}=f(y,x)$ since multiplication is commutative. –  robjohn Mar 2 '13 at 15:18
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