- #1
jostpuur
- 2,116
- 19
Suppose [itex]\mathfrak{g}[/itex] and [itex]\mathfrak{h}[/itex] are some Lie algebras, and [itex]G=\exp(\mathfrak{g})[/itex] and [itex]H=\exp(\mathfrak{h})[/itex] are Lie groups. If
[tex]
\phi:\mathfrak{g}\to\mathfrak{h}
[/tex]
is a Lie algebra homomorphism, and if [itex]\Phi[/itex] is defined as follows:
[tex]
\Phi:G\to H,\quad \Phi(\exp(A))=\exp(\phi(A))
[/tex]
will [itex]\Phi[/itex] be a group homomorphism?
Since [itex]\exp(A)\exp(B)=\exp(A+B)[/itex] is not true in general, I see no obvious way to prove the claim.
[tex]
\phi:\mathfrak{g}\to\mathfrak{h}
[/tex]
is a Lie algebra homomorphism, and if [itex]\Phi[/itex] is defined as follows:
[tex]
\Phi:G\to H,\quad \Phi(\exp(A))=\exp(\phi(A))
[/tex]
will [itex]\Phi[/itex] be a group homomorphism?
Since [itex]\exp(A)\exp(B)=\exp(A+B)[/itex] is not true in general, I see no obvious way to prove the claim.