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(adsbygoogle = window.adsbygoogle || []).push({});

[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.

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# Generating group homomorphisms between Lie groups

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