If $\mathfrak{g}$ is a real or complex [[Lie algebras]], then the finite dimensional complex [[Representation]] is a given as: $\pi : \mathfrak{g}\rightarrow \mbox{gl}{(\mathcal{V})}$ Here $\mbox{gl}{(\mathcal{V})}$ is the space of all linear operators ([Linear map](Linear%20map.md)), that are denoted as $\mathcal{V} \rightarrow \mathcal{V}$ under a [Lie bracket](Lie%20bracket.md), $[X,Y]$. %%I'm not sure this makes sense. This is too vague.%% # Emergence from [[Lie group representations]]s Where we have a [simply connected Lie group](Simply%20connected%20Lie%20groups.md), $\mbox{G}$ and we consider $\Pi$ to be a _representation of $\mbox{G}$ acting on $\mathcal{V}$_ we express this [Lie group representations](Lie%20group%20representations.md) as:  which, through the [matrix exponential](Matrix%20exponentials.md#Algebraic%20properties%20matrix%20exponentials), relates to the unique representation of $\pi(X)$ of $\mathfrak{g}$. $\pi(X)$ can be computed $\forall X$ from $\Pi$ via the expression, $\pi(X) = \frac{d}{dt}\Pi(e^{tX})|_{t=0}$ ^1514ad which satisfies $\pi(AXA^{-1}) = \Pi(A)\pi(X)\Pi(A)^{-1}$. ^7a8aad #MathematicalFoundations/Algebra/AbstractAlgebra/GroupTheory/Lie/LieGroups/Algebras/LieAlgebras #MathematicalFoundations/Algebra/AbstractAlgebra/RepresentationTheory