Is the Lie Product Always Unique in Vector Spaces?

[Rasalhague]

Are there cases where there's more than one binary operation to choose from by which to define a Lie algebra for a given vector space?

 

[Rasalhague]

First guess, yes. (Unless, as often, there are multiple definitions in play.)

"The [sic] Lie algebra of the vector space Rⁿ is just Rⁿ with the Lie bracket given by [A,B] = 0" (Wikpedia: Lie group).

"The three-dimensional Euclidean space R³ with the Lie bracket given by the cross product of vectors becomes a three-dimensional Lie algebra" (Wikipedia: Lie algebra).

 

[Fredrik]

If you're asking if there exists a vector space with two different Lie brackets, the answer is definitely yes. There are even two Lie brackets on the tangent space at the identity of a Lie group, one constructed using left multiplication and the other using right multiplication. (These two Lie algebras are isomorphic, so it doesn't matter which one we call "the" Lie algebra of the Lie group).

If you're looking for an example of a vector space with two non-trivial Lie brackets that give us non-isomorphic Lie algbras, I don't have one, but I would be surprised if no such example exists. (By "non-trivial", I mean that it's not the bracket defined by [X,Y]=0 for all X,Y).

 

[Rasalhague]

Thanks. That's exactly what I wanted to know.