= Exceptional isomorphism between sp4 and so5
{title2=$\mathfrak{sp}_4\cong\mathfrak{so}_5$}
Over $\mathbb C$, the <symplectic Lie algebra> on a four-dimensional <symplectic vector space> acts faithfully on its five-dimensional <primitive exterior square>, preserving its <symmetric bilinear form>. Faithfulness follows because an operator killing all wedges $e_i\wedge e_j$ has all off-diagonal coefficients zero and satisfies $A_{ii}+A_{jj}=0$ for every pair, hence is zero. The source and target <Lie algebras> both have <dimension> ten, so this faithful map is an <isomorphism> to the <Special orthogonal Lie algebra> $\mathfrak{so}_5$.
Back to article page
