= Defining representation of a matrix Lie algebra
{title2=$X\cdot v=Xv$}
= Defining representation
{synonym}
For a <Lie subalgebra> $\mathfrak g\subseteq\mathfrak{gl}(V)$, its defining representation is its given action on the <vector space> $V$. The <Lie bracket> identity $[X,Y]v=X(Yv)-Y(Xv)$ verifies the representation property. For the <special linear Lie algebra>, diagonal trace-zero matrices have coordinate <weights of a representation> $L_i$ on the standard basis; for a <Special orthogonal Lie algebra>, the same action preserves the defining symmetric <bilinear form>.
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