If $\mathfrak{g}$ and $\mathfrak{h}$ are both [Lie algebras](Lie%20algebras.md), a linear map $\phi:\mathfrak{g}\rightarrow\mathfrak{h}$ is a Lie algebra homomorphism if $\phi([X,Y]) = [\phi(X),\phi(Y)]$, for all $X,Y \in \mathfrak{g}$ where $[.,.]$ denotes a [Lie bracket.](Lie%20algebras.md#^73b451) # Emergence of Lie Algebra homomorphisms from Lie Group homomorphisms Given a [Lie group homomorphism](Lie%20group%20homomorphisms.md), $\Phi:\mbox{G}\rightarrow \mbox{H}$, these groups share a unique real linear [linear map](Linear%20map.md), $\phi:\mathfrak{g}\rightarrow\mathfrak{h}$ such that $\Phi(e^X)=e^{\phi(X)}\;\; X\in \mathfrak{g}$ ([proof](Proof%20of%20the%20emergence%20of%20Lie%20algebra%20homomorphisms%20from%20Lie%20group%20homomorphisms.md)). This map _is a Lie Algebra homomorphism_ that has the following properties [listed here.](Lie%20algebra%20homomorphisms.md#Properties%20of%20Lie%20algebra%20homomorphisms) ^aa26c2 ## Properties of Lie algebra homomorphisms Given a Lie group homomorphism, $\Phi$ and a its corresponding Lie algebra homomorphism $\phi,$ the properties [of Lie algebra homomorphisms in relation to Lie group homomorphisms](Lie%20algebra%20homomorphisms.md#Emergence%20of%20Lie%20Algebra%20homomorphisms%20from%20Lie%20Group%20homomorphisms) are as follows     --- # Proofs and Examples ## Proof of the [emergence of Lie algebra homomorphisms from Lie group homomorphisms](Lie%20algebra%20homomorphisms.md#Emergence%20of%20Lie%20Algebra%20homomorphisms%20from%20Lie%20Group%20homomorphisms)     ## Proofs of [properties of Lie algebra homomorphisms in relation to Lie group homomorphisms](Lie%20algebra%20homomorphisms.md#Properties%20of%20Lie%20algebra%20homomorphisms) Given a [Lie group homomorphism](Lie%20group%20homomorphisms.md), $\Phi$ and a its corresponding Lie algebra homomorphism $\phi,$ along with a free parameter $t\in\mathbb{R},$ we prove the [properties of Lie algebra homomorphisms](Lie%20algebra%20homomorphisms.md#Properties%20of%20Lie%20algebra%20homomorphisms) below:    #MathematicalFoundations/Algebra/AbstractAlgebra/GroupTheory/Lie/LieGroups/Algebras/LieAlgebras #MathematicalFoundations/Algebra/AbstractAlgebra/GroupTheory/Lie/LieGroups