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Super Quick Question

  1. Nov 13, 2005 #1
    When dealing with one-forms, what is the property:
    dx/\dy = -dy/\dx
    formally called? Super-commutativity? Anti-commutativity?
     
  2. jcsd
  3. Nov 13, 2005 #2

    HallsofIvy

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    Anti-commutativity.
     
  4. Nov 13, 2005 #3

    mathwonk

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    technically and commonly true, but oddly, I think sometimes it is just called commutativity. i.e. the rule that links the sign of a permutation with the degree of the form is so common, that obeying it is sometimes called commutativity.
    i.e. in this case we are looking at one forms, so we have anticommutativity, but when you permute two "2-forms" there is no sign change.
    so when you look at the behavior of this multiplication on forms of all degrees, not just those of odd degree, the rule is not as simple to describe by calling it "anticommutativity".
    so then I think it is called something else, like commutativity or degree commutativity, or something that escapes me.
    of course this is confusing since it is not actual commutativity, but there is some convention for distinguishing cases where there is a rule for commutativity, from where there is no rule at all.
    so there are two simple rules for how multiplication behaves under permuting the factors by s, the simplest being true commutativity, where the product changes by a factor of +1, i.e. not at all, and the other simple rule is where a homogeneous product changes by a factor of [(-1)^s]d, the sign of the permutation raised to the degree of the factors.
    in a sense either of these is so simple and expected that it is considered a type of commutativity.
    i have encountered this in algebraic topology in the case of cup product of cohomology, i.e. differential forms if you use de-rham cohomology.
    e.g. see roughly page 215 of allen hatcher's book, algebgraic topology, below the discussion of theorem 3.14.
    here is a little exercise for you: prove there are essentially no other possible "commutativity" conventions besides these two.
    i.e. classify group homomorphisms from the group S(n) to the multiplicative group say, of non zero complex numbers.
     
    Last edited: Nov 13, 2005
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