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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1By 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. – Feb 05 '12 at 11:59
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3I 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 05 '12 at 13:55
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I really like $[x,y]_f$. – Peteris Feb 05 '12 at 21:52