Exploring Lie Algebras: Questions about Adjoint Maps

[Pietjuh]

I've been studying some things about Lie algebras and I've got some questions about it.

We know that for a given Lie group G, the tangent space at the identity has the structure of a Lie algebra, where the Lie bracket is given by [X,Y]=ad(X)(Y). This map ad is the differential at e of the map Ad which is defined by Ad(g)(X) = gXg^{-1}. Now I'm wondering whether it's possible to associate a Lie algebra structure on tangent spaces at points near the identity element, for example in an open set containing e, or for points in the same connected component in which e lies. I don't think you could use the map Ad for it, but perhaps some slight modification is possible??

Thanks in advance! :)

 

[Cexy]

I don't see an immediate problem - rather than taking the definition of the Lie algebra L(G) associated with the group G to be the tangent space at the identity, just take it to be the algebra of vector fields on G. Then any vector field $$V_X(g)$$ which is generated by applying the push-forward $$L_{g*}$$ to the vector $$X \in T_eG$$ can equally well be generated by first applying $$L_{p^{-1}*}$$ to a vector $$Y \in T_pG$$ and then applying $$L_{g*}$$.

 

[tacman]

First of all, it's great to hear that you're exploring Lie algebras! They are a fascinating topic in mathematics and have many applications in various fields. Regarding your questions about adjoint maps, let me try to provide some clarification.

You are correct in thinking that the map Ad cannot be used to define a Lie algebra structure on tangent spaces at points near the identity. This is because the definition of Ad requires a group element g, and points near the identity may not necessarily have a corresponding group element.

However, there is a way to extend the Lie algebra structure to tangent spaces at points near the identity, using the concept of left-invariant vector fields. These are vector fields on the Lie group that are invariant under left translations by group elements. The set of all left-invariant vector fields forms a Lie algebra, known as the Lie algebra of left-invariant vector fields.

Now, using the exponential map, which is a map from the Lie algebra to the Lie group, we can "push forward" this Lie algebra structure to tangent spaces at points near the identity. This is done by taking the tangent space at a point g and identifying it with the tangent space at the identity using the left translation by g.

In summary, while the map Ad cannot be used to define a Lie algebra structure on tangent spaces at points near the identity, we can use the concept of left-invariant vector fields and the exponential map to extend the Lie algebra structure. I hope this helps clarify your doubts. Keep exploring and asking questions, and you'll continue to deepen your understanding of Lie algebras. Good luck!