= Invariant vector of a Lie algebra representation
{title2=$V^{\mathfrak g}=\bigcap_X\ker\rho(X)$}
= Invariant vectors of a Lie algebra representation
{synonym}
For a <Lie algebra representation> $\rho:\mathfrak g\to\mathfrak{gl}(V)$, an invariant vector is a vector $v$ annihilated by every represented Lie algebra element:
$$
V^{\mathfrak g}=\{v\in V:\rho(X)v=0\text{ for all }X\in\mathfrak g\}=\bigcap_{X\in\mathfrak g}\ker\rho(X).
$$
The condition is zero infinitesimal action, not $\rho(X)v=v$. The invariant vectors form a <vector subspace>.
For a one-dimensional abelian Lie algebra acting on $K^2$ through $\rho(X)=\operatorname{diag}(0,1)$, the invariant vectors are exactly the first coordinate line. For the <Adjoint representation>, invariants are the <center of a Lie algebra>, because $\rho(X)v=[X,v]$.
If the representation differentiates a representation of a connected <Lie group>, Lie algebra invariants are also group invariants. Connectedness matters: a disconnected group can act nontrivially even when its Lie algebra acts trivially. The sign action of $\{1,-1\}$ has zero Lie algebra but no nonzero fixed vector.
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