Uniqueness of Lie product

[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.

 

[blue_raver22]

The Lie product, also known as the Lie bracket, is a fundamental operation in the field of Lie algebras. It plays a crucial role in defining the structure and properties of Lie algebras, which are mathematical objects used to study the symmetry of continuous systems. The uniqueness of the Lie product is a concept that is often discussed in the context of Lie algebras.

To answer the question, it is important to first understand what is meant by the uniqueness of the Lie product. The uniqueness of the Lie product refers to the fact that for a given vector space, there is only one binary operation that can be used to define a Lie algebra. This operation must satisfy certain properties, such as being bilinear, antisymmetric, and satisfying the Jacobi identity.

In most cases, the Lie product is uniquely determined by the structure of the vector space and the properties it must satisfy. However, there are some special cases where there may be more than one binary operation that can be used to define a Lie algebra for a given vector space.

One such case is when the vector space is infinite-dimensional. In this case, there may be multiple choices for the Lie product, each of which satisfies the required properties. This is due to the fact that infinite-dimensional vector spaces have more degrees of freedom, allowing for more possibilities for the Lie product.

Another case where there may be more than one Lie product is when the vector space has a non-trivial topology. In this case, the choice of Lie product may depend on the topology of the space, leading to multiple options for defining the Lie algebra.

It is important to note that even in cases where there may be more than one choice for the Lie product, all of these choices will result in isomorphic Lie algebras. This means that while the Lie algebras may be defined by different binary operations, they will still have the same structure and properties, and can be considered equivalent.

In conclusion, while the uniqueness of the Lie product is a general concept, there are some cases where there may be more than one binary operation that can be used to define a Lie algebra for a given vector space. However, all of these choices will result in isomorphic Lie algebras, and will not affect the overall study and understanding of Lie algebras.