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Lie algebra

  1. Apr 28, 2006 #1
    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! :)
  2. jcsd
  3. Apr 29, 2006 #2
    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 [tex]V_X(g)[/tex] which is generated by applying the push-forward [tex]L_{g*}[/tex] to the vector [tex]X \in T_eG[/tex] can equally well be generated by first applying [tex]L_{p^{-1}*}[/tex] to a vector [tex]Y \in T_pG[/tex] and then applying [tex]L_{g*}[/tex].
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