Given [matrix Lie group](Matrix%20Lie%20group.md)s $\mbox{G}$ and $\mbox{H}$, a map, $\Phi$ between these groups is a _Lie group homomorphism_ if: 1) $\Phi$ is a group homomorphism and 2) $\Phi$ is continuous. ^a80ddb Continuity for a matrix Lie Group is almost always a given. An example of a Lie group homomorphism is the [determinant](Determinants.md). %%Is this still true for abstract Lie Groups? Are Lie group homomorphisms also representations?%% # Emergence of Lie Algebra homomorphisms from Lie Group homomorphisms  ## Properties of Lie group homomorphisms Given a Lie group homomorphism, $\Phi$ and a its corresponding Lie algebra homomorphism $\phi,$ the properites of [Lie group homomorphisms in relation to Lie algebra homomorphisms](Lie%20group%20homomorphisms.md#Emergence%20of%20Lie%20Algebra%20homomorphisms%20from%20Lie%20Group%20homomorphisms) has the following properties 1) $\phi(AXA^{-1})=\Phi(A)\phi(X)\Phi(A)^{-1}$ for all $X\in \mathfrak{g}$ and $A \in \mbox{G}$. ^80fb3a 2) $\phi([X,Y]) = [\phi(X),\phi(Y)]$ for all $X,Y \in \mathfrak{g}$. ^30de39 3) $\phi(X) =\frac{d}{dt}\Phi(e^{tX})|_{t=0}$ for all $X \in \mathfrak{g}$. ^ecc337 These properties are proven ([here](Lie%20group%20homomorphisms.md#Proofs%20of%20properties%20of%20Lie%20group%20homomorphisms%20in%20relation%20to%20Lie%20algebra%20homomorphisms%20Lie%2020group%2020homomorphisms%20md%20Properties%2020of%2020Lie%2020group%2020homomorphisms)). A [Lie group homomorphism](Lie%20group%20homomorphisms.md) always gives rise to a Lie Algebra homomorphism because of property 2. This also means that the derivative is a Lie Algebra homomorphism. ^883e05 --- # Proofs and examples ## Proof of emergence of Lie algebra homomorphisms from Lie Group homomorphisms     ## Proofs of [properties of Lie group homomorphisms in relation to Lie algebra homomorphisms](Lie%20group%20homomorphisms.md#Properties%20of%20Lie%20group%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 group homomorphisms](Lie%20group%20homomorphisms.md#Properties%20of%20Lie%20group%20homomorphisms) as follows:    #MathematicalFoundations/Algebra/AbstractAlgebra/GroupTheory/Lie/LieGroups #MathematicalFoundations/Algebra/AbstractAlgebra/GroupTheory/Lie/LieGroups/Algebras/LieAlgebras