If $\Pi$ is a finite dimensional [[Lie group representations]] of $\mbox{G}$ acting on a [space](Vector%20spaces.md) $\mathcal{V}$, A [subspace](Vector%20spaces.md#Subspaces%20of%20vector%20spaces) $\mathcal{W} \subset \mathcal{V}$ is _invariant_ under actions of $\mbox{G}$ if $\Pi(A)w \in \mathcal{W}$ for all $w\in\mathcal{W}$ and all $A\in \mbox{G}$. If $\mathcal{W}\neq\mathcal{V}$ and $\mathcal{W}\neq\{0\}$ then $\mathcal{W}$ is nontrivial. And if the representation has no non-trivial [invariant subspaces](Invariant%20subspaces.md) then it is _irreducible_. If there are non-[trivial](Invariant%20subspaces.md#The%20trivial%20invariant%20subspace) [invariant subspaces](Invariant%20subspaces.md), then the representation is _reducible_. %%Not clear if trivial here refers only to the $\{0\}$ subspace. %% # Irreducible Lie algebra representation This definition is the same for [[Lie algebra representation]]s, $\pi$. If $\mbox{G}$ is a [connected](Connected.md) [matrix Lie group](Matrix%20Lie%20group.md) with [Lie algebra](Lie%20algebras.md) $\mathfrak{g}$ where $\Pi$ and $\pi$ are the respective representations of the group and algebra, then $\Pi$ is _irreducible_ if and only if $\pi$ is _irreducible_. #MathematicalFoundations/Algebra/AbstractAlgebra/GroupTheory/Lie/LieGroups #MathematicalFoundations/Algebra/AbstractAlgebra/GroupTheory/Lie/LieGroups/Algebras/LieAlgebras #MathematicalFoundations/Algebra/AbstractAlgebra/RepresentationTheory