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811:. This discussion is clearly from Liboff, I have a copy and I have located the subject matter in the text, but my copy is the 2nd edition, not the 4th edition listed in the References section of the article. Would someone who has access to the 4th edition please update the page number of the reference? —
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In the first section under Group Theory, it is stated that "Note that one must consider the subgroup generated by the set of commutators because in general the set of commutators is not closed under the group operation." Why does closure not exist? The operation is a binary operation, the elements
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The section on ring identities is strange and confusing, in that it launches straight into Lie algebra identities, without developing any other results first. Where are the finite rings? Why jump straight to algebras? Maybe this section needs a new heading? Perhaps more identities can be found
1102:, is either (a) completely wrong, or at very best (b) out of place and with missing assumptions. As to why it's considered acceptable to remove a flag for attention without getting due clarification first, I'm completely in the dark.
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The third property is redundant because and = 0 imply it. This is according to
Abraham and Marsden's Foundations of Mechanics. Should we get rid of property 2 or 3, or put it in under the commutator identities?
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is closed under inverses, for = , and so the inverse of any commutator is again a commutator. However, in general the product of two commutators need not be a commutator. The canonical example is the
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chosen come from the group. And in fact, it is a group we are talking about from which commutators are being generated. By definition of a group, we should have closure.
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I don't know what you are referring to. The identity = A + B is nowhere in the article, but = C + B and = A + B are in the article, and both are correct.—
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I suggest that we add a brief section or one-liner about the anticommutator: {A,B} := AB + BA since this notation is not mentioned in the
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You were right about number 5. Thank you for pointing this out. Number 6 was also incorrect. I have removed them.—
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if and only if it is closed under inverses and products, i.e., if the following two statements are true:
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In the Ring theory section, mention is made of the connection between the commutator, the
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are the matrices of the same two operators (in the same order) with respect to another basis, then
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on
Knowledge. If you would like to participate, please visit the project page, where you can join
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The problem is that in general the product of two commutators need not be a commutator.
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In linear algebra, if two matrices commute in one basis they will commute in any basis.
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Nevermind. I just looked up a copy in the library and fixed this myself. —
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redirects here). I made a quick start, feel free to flesh it out a bit. --
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article and comes up a lot (at least in physics). Cheers
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