/sci/ - Science & Math

>>12423727
you sound like you don't know what a group is

>>12423727
>why cant you just list the group multiplication rules to define it
the commutators do precisely that, you retard. Once you stated all the commutators of all the generators, your algebra is well-defined.

>>12424153
I asked why do you need to state that information with commutators and not just by explicitly giving the group action, as is done for quaternions.
Eg: i*i=-1.
So why is this information put in the form of commutators?

>>12424060
Sneed

An infinitesimal group ("Lie algebra") is a linearization of a transformation group ("Lie group")

>>12424197
I mean, if if knew the direct multiplication rules it would be trivial to write down tge commutators:
A*B=C
B*A=D
So AB-BA=C-D or [A,B]=C-D
And thats ok, but why define the group multiplication with these brackets while you can just say AB=C, BA=D? What's so particular about commutators?

>>12424203
And water is wet, whats your point?

>>12424214
not every lie bracket is a commutator in some group

>>12424225
Can you just write down a table of nxn equations defining the group binary operation gxg->g without having to talk about commutators?

>>12423727
The structure constant is the (real part of the) scalar which multiplies the group element on the right side of a commutator equality. Sometimes it is trivial, as it is for the quaternions, but sometimes the structure constant is more complicated.

Bump

>>12424254
in a general lie algebra there's no group associated with it, no group binary operation, no commutator. only the bracket. it just turns out that in the special case of a lie algebra of a matrix lie group, the bracket is exactly the commutator taken in the ring of matrices. your question is meaningless for general lie algebras. so if you want to use general theory of lie algebras, you need to use brackets.

>>12423727
>>12424254
This guy >>12424153 was right, but he understated his point and I think you're confused to "what a Lie algebra" is. The definition of a Lie algebra, to quote the first line of the Wikipedia article is
>In mathematics, a Lie algebra is a vector space together with an operation called the Lie bracket, an alternating bilinear map, that satisfies the Jacobi identity.
The point is that the given operation - and indeed the only given operation! - is the Lie bracket. I.e. the "Lie bracket" [A,B] is apriori not A·B-B·A.

Here's a clip on the Axioms
https://youtu.be/cCbn5x0z7pA

There are then rather technical result that savior the situation, in that it turns out that there's isomorphisms between Lie algebras and algebras with multiplications "·" that indeed makes it true that Lie algebras are "of the form A·B-B·A".
https://en.wikipedia.org/wiki/Poincar%C3%A9%E2%80%93Birkhoff%E2%80%93Witt_theorem

I now see the statement is a burried in the article as a corollary.

Btw. a similar blunder is to think of Lie groups as always representable by finite dimensional linear operators.