For a [matrix Lie group](Matrix%20Lie%20group.md) $\mbox{G}$, its representation is a [representation of a finite group](Group%20representation.md#Representation%20of%20a%20Finite%20Group). Where we have a Lie Group and we consider $\Pi$ to be a _representation of $\mbox{G}$ acting on a complex vector space $\mathcal{V}$_, we may write $\Pi(e^{X}) = e^{\pi(X)}$ ^7f4204 or $\Pi(e^{tX}) = \Pi(g)= e^{\pi(X)},$ which, with the [matrix exponential](Matrix%20exponentials.md#Role%20of%20matrix%20exponentials%20in%20Lie%20groups%20and%20Lie%20algebras), forms a unique representation of $\pi(X)$ of $\mathfrak{g}$ where $g\in\mbox{G}$ and the representation is the linear action on $\mathcal{V}$, $g\cdot\mathcal{V}$. # Corresponding [Lie algebra representation](Lie%20algebra%20representation.md) The [related Lie Algebra representation](Lie%20algebra%20representation.md#Emergence%20from%20Lie%20group%20representations%20s), $\pi(X),$ can be computed for all $X$ from $\Pi$ via the expression,   It is NOT necessarily true that every [[Lie algebra representation]] $\pi$ comes from a corresponding Lie Group Representation, $\Pi$. However this is true if $\mbox{G}$ is a [simply connected Lie group.](Simply%20connected%20Lie%20groups.md) #MathematicalFoundations/Algebra/AbstractAlgebra/GroupTheory/Lie/LieGroups #MathematicalFoundations/Algebra/AbstractAlgebra/RepresentationTheory