Talk:Lie algebra

Category theoretic definition

The definition in that section looks a bit useless and seems to say that category theory complicates things. Perhaps the category of vector spaces over k should be replaced by any braided monoidal k-linear category. The notion of a Lie algebra can be defined (and is studied) in this generality, and I guess that this is what the category theoretic definition is all about. Martin B. 92.231.13.68 (talk) 18:30, 28 May 2015 (UTC)

Well, apparently, someone removed this at some point. It's a shame, because the category-theoretic definition helps build insight and understanding into what is going on; once you understand Lie algebras, the cat-th defition helps you go say "a hah!" to the next level. Anyway, some other comment, up above (in Archive 1 of this talk page) noted:

The category theoretic definition provided doesn't work over a field of characteristic 2. The correct definition is as follows:""

A Lie algebra is an object A in the category of vector spaces together with a morphism $[\cdot ,\cdot ]:A\otimes A\rightarrow A$ such that

$[\cdot ,\cdot ]\circ \Delta =0$ where $\Delta :A\rightarrow A\otimes A$ is the diagonal morphism, and

$[\cdot ,\cdot ]\circ ([\cdot ,\cdot ]\otimes id)\circ (id+\sigma +\sigma ^{2})=0$ where σ is the cyclic permutation braiding $(id\otimes \tau _{A,A})\circ (\tau _{A,A}\otimes id)$.

This is not correct since the diagonal "morphism" doesn't exist in the category of vector spaces. — Preceding unsigned comment added by 79.155.213.93 (talk) 17:39, 10 March 2021 (UTC)

(It should probably also be added that the morphism $\tau _{A,A}:A\otimes A\rightarrow A\otimes A$ is the interchange morphism rather than leave that as assumed knowledge.)

This content should be restored and expanded. 67.198.37.16 (talk) 20:03, 31 October 2020 (UTC)

I restored it. I did not expand it. 67.198.37.16 (talk) 20:42, 31 October 2020 (UTC)

This is a rather old thread. However, Maricumpli has removed this section again, because the "diagonal morphism" that was used (and is used in the above quotation) is not a morphism. This is true. I have not found how to fix this error by using alternativity. However, it can be fixed by using antisymmetry, which excludes the characteristic two. So I have rewritten the section this way, and, by the way, I have improved the style (MOS:MATH). D.Lazard (talk) 18:03, 10 March 2021 (UTC)

I agree and think the first response that it provides "insight" is unjustified. It's nothing more than a rephrasing of the definition that does not provide any insight. This section should be deleted.

Anticommutativity follows from what?

I am not familiar with Lie algebras, but I don't see how the anticommutativity follows from the bilinearity and alternativity properties alone, without the use of the Jacobi relation. Could a hint to why it does be added? (Or could it be added that the Jacobi relation is also necessary).

When I show anticommutativity (using all three relations), I assume that the Lie bracket with any argument z and the zero element (0) of the vector space is 0, but I am not sure that this is true.

To whoever made an edit, thank you! A few more words (not even one line) and the article has probably been made clearer to many readers. — Preceding unsigned comment added by 130.225.98.190 (talk) 09:48, 27 August 2015 (UTC)

Once upon a time (maybe it was removed??) there was an explicit discussion of this. Basically, if you try to write out every possible way that an associative algebra could work, you are forced into having only two choices. Bilinearity allows only one of these two possibilities. There are similar arguments that can be made in various other kinds of algebras (not just Lie algebras) and I've noticed that, more than once, that some of these were removed; one edit summary called them "pointless philosophical ruminations". Of course, as your question shows, they are not pointless at all. I reverted that one, but I can't go run around and try to police these kinds of (destructive) edits. Perhaps someone else can find/reconstruct this argument; it can be found in the better-written textbooks. 67.198.37.16 (talk) 19:43, 31 October 2020 (UTC)

Merge proposal for Classical Lie algebras

It has been proposed that classical Lie algebras be merged here.

• Oppose: I feel that they warrant an article of their own, much like classical group displays the unique status (application and mathematics-wise) of these groups. On the technical side, the classical Lie algebras can be treated in a surprisingly uniform manner. YohanN7 (talk) 08:36, 5 May 2017 (UTC)
• Oppose: Per YohanN7. There is a lot of activity w.r.t affine/non-rigid stuff, and so having a distinct article focused on the classical cases make sense, this can avoid having to jam everything into one article. 67.198.37.16 (talk) 19:31, 31 October 2020 (UTC)

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Rotation Velocity Concern

At the end of the intro, we find ...

This is the Lie algebra of the Lie group of rotations of space, and each vector $v\in \mathbb {R} ^{3}$ may be pictured as an infinitesimal rotation around the axis $v$, with velocity equal to the magnitude of $v$.

My concern is that the magnitude of $v$ represents the angle (or magnitude) of the rotation, not its velocity.

Am not confident enough in my concern to edit the page, hence am adding this topic, here.

Any thoughts? WadoNeil (talk) 10:24, 12 April 2023 (UTC)

As the velocity mentioned at the end of the sentence is not related to anything, I think that the module of $v$ might stand for the angle of rotation than for the velocity. Stefan Groote (talk) 05:01, 13 April 2023 (UTC)