# Talk:Commutator

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WikiProject Mathematics (Rated Start-class, Mid-priority)

## Matrices in bases

What does this statement mean?

In linear algebra, if two matrices commute in one basis they will commute in any basis.

It seems to be trying to mean that, if ${\displaystyle A}$ and ${\displaystyle B}$ are the matrices of two linear operators with respect to one basis, and ${\displaystyle A'}$ and ${\displaystyle B'}$ are the matrices of the same two operators (in the same order) with respect to another basis, then ${\displaystyle A}$ and ${\displaystyle B}$ commute if and only if ${\displaystyle A'}$ and ${\displaystyle B'}$ do. However, the statement as it stands is meaningless. JadeNB 21:18, 10 October 2006 (UTC)

## Standard notation

The standard mathematical notation for commutator is ${\displaystyle \scriptstyle {ad(x)(y)=[x,y]}}$. As I see no reason to depart from the standard, I changed the relevant paragraph. AlainD 10:02, 25 January 2007 (UTC)

I think it's misleading to use the notation "ad(x)^3(y) = [x,[x,[x,y]]]", since there are a lot of different algebras floating around. In particular, ad(x) is a derivation, which has both the usual algebra structure of composition, and a Lie algebra structure induced from the former. Of course y lives in a Lie algebra, where something like y^3 is not defined. I admit that ad(x) may be unambiguous mathematically, but I feel that in this context it's best to avoid confusion if possible. Tesseran 07:46, 28 June 2007 (UTC)

## Brief section about the anticommutator

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 anticommutativity article and comes up a lot (at least in physics). Cheers A13ean (talk) 22:50, 5 October 2008 (UTC)

Agreed (especially since anticommutator redirects here). I made a quick start, feel free to flesh it out a bit. --Strange but untrue (talk) 17:34, 10 November 2008 (UTC)

## Reference needed for uncertainty principle

In the Ring theory section, mention is made of the connection between the commutator, the Heisenberg uncertainty principle and the Robertson-Schrödinger relation. 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? — Anita5192 (talk) 05:13, 25 March 2012 (UTC)

Nevermind. I just looked up a copy in the library and fixed this myself. — Anita5192 (talk) 04:51, 27 July 2012 (UTC)

## Ring commutator

Should Ring commutator redirect here? — Preceding unsigned comment added by 70.247.173.205 (talk) 04:56, 1 March 2016 (UTC)

## Ring identities

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 for these cases? — Preceding unsigned comment added by 70.247.173.205 (talk) 05:00, 1 March 2016 (UTC)

## Non-closure?

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 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. 50.35.103.217 (talk) 22:46, 15 September 2017 (UTC)

The problem is that in general the product of two commutators need not be a commutator.
Let G be a group under a given binary operation ∗. By definition, a non-empty subset H of a group G is a subgroup of G if and only if it is closed under inverses and products, i.e., if the following two statements are true:
1. The inverse a−1 with respect to ∗ of any element a of H is again in H.
2. The product ab of any two elements of H is again in H.
Now let H be the set of commutators of elements of G. This subset of G is closed under inverses, for [ab]−1 = [ba], 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 free group over the four-element set {a, b, c, d}. Here the element [ab][cd] cannot be written as a commutator. – Tea2min (talk) 18:28, 16 September 2017 (UTC)

## Ring Theory Lie Algebra Identities

The third property is redundant because [A + B, C] and [A, A] = 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?

MichalKononenko (talk) 05:18, 18 January 2018 (UTC)

## Lie-Algebra identities

Property no. 4 in the Lie Algebra identities (https://wiki.alquds.edu/?query=Commutator#Lie-algebra_identities) is either plain wrong and inconsistent with, say, the initial identity under Additional Identities (i.e. [AB,C] = A[B,C]+B[A,C]) or it requires a much better explanation of why and when it holds.

Episanty (talk) 21:07, 18 August 2018 (UTC)

I don't know what you are referring to. The identity [AB,C] = A[B,C] + B[A,C] is nowhere in the article, but [A,BC] = [A,B]C + B[A,C] and [AB,C] = A[B,C] + [A,C]B are in the article, and both are correct.—Anita5192 (talk) 00:22, 19 August 2018 (UTC)
The identity [AX+BY,Z]=A[X,Z]+B[Y,Z], currently numbered no. 5 at https://wiki.alquds.edu/?query=Commutator#Lie-algebra_identities, 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. Episanty (talk) 18:03, 20 August 2018 (UTC)
You were right about number 5. Thank you for pointing this out. Number 6 was also incorrect. I have removed them.—Anita5192 (talk) 21:23, 20 August 2018 (UTC)
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