Talk:Commutator

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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)Reply

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)Reply

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)Reply

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)Reply

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

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

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)Reply

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)Reply

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 a∗b 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 [a, b]⁻¹ = [b, a], 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 [a, b][c, d] cannot be written as a commutator. – Tea2min (talk) 18:28, 16 September 2017 (UTC)Reply

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)Reply

Property no. 4 in the Lie Algebra identities (https://en.wikipedia.org/wiki/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)Reply

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)Reply

The identity [AX+BY,Z]=A[X,Z]+B[Y,Z], currently numbered no. 5 at https://en.wikipedia.org/wiki/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)Reply

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)Reply

I feel like the commutator of integer powers of ring elements formula is wrong. It especially doesn't seem to agree with the identities previously presented, such as the 3rd one for example. As there is no citation of the formula for the commutator of integer powers I couldn't check it myself. Maybe someone with more background than me in maths can check if the formula is valid. Juliq-g (talk) 17:36, 11 March 2025 (UTC)Reply

The article clearly states that there are two conventions - the first is [g, h] = g⁻¹h⁻¹gh and the second is [g, h] = ghg⁻¹h⁻¹. The article uses the first, but only gives references for the second... So we could improve the article by (1) giving references for the first convention and (2) explaining why mathematicians prefer it over the second. Sam nead (talk) 10:11, 7 April 2025 (UTC)Reply

For what it's worth, every book on group theory that I have ever seen uses the first convention. The article was started (in 2001) using that convention, without giving any references. Perhaps at the time it wasn't clear that the convention used actually needed references to support it, but that's speculation. Later versions of the article (around 2010) used to say

N.B. The above definition of the commutator is used by group theorists. Many other mathematicians define the commutator as

[g, h] = ghg⁻¹h⁻¹.

This formulation was then changed a few times.

It's always hard to find references stating that some convention is more common than another convention, and conventions sometimes change over time. I couldn't say why one convention would be more common than the other. Many (introductory) books mention commutators only as part of the exercises without any further elaboration. Anyway, I own the following books that use the first convention:

• Aschbacher, Michael (2000). Finite group theory. Cambridge University Press. p. 26. ISBN 0-521-78675-4.
• Rose, John S. (2012) [unabridged and unaltered republication of a work first published by the Cambridge University Press, Cambridge, England, in 1978]. A Course on Group Theory. Dover Publications. p. 58. ISBN 978-0-486-68194-8.
• Isaacs, I. Martin (1994) [unabridged, corrected republication of the work first published by Academic Press, New York in 1976]. Character Theory of Finite Groups. Dover Publications. p. 45. ISBN 978-0486680149.
• Dixon, John D. "5. Commutators and Derived Series". Problems in Group Theory. p. 30. ISBN 0-486-45916-0.

By contrast,

• Pinter, Charles C. A Book of Abstract Algebra (2nd ed.). p. 152. ISBN 978-0-486-47417-5.

uses the second convention. – Tea2min (talk) 14:11, 7 April 2025 (UTC)Reply

To add to this: Herstein (1975) Topics in algebra, page 252 uses the first convention.