= Contraction splitting of a defining module tensor a dual symmetric square
{title2=$V\otimes\operatorname{Sym}^2V^*=\ker C\oplus\iota(V^*)$}
For the defining $n$-dimensional <special linear Lie algebra> module, with $n\geq2$, write $T_i(p)=e_i\otimes p(y_1,\ldots,y_n)$. The equivariant contraction is $C(T_i(p))=\partial_{y_i}p$. The map $\iota(f)=\sum_i e_i\otimes y_i f$ is also equivariant, since $\sum_i e_i\otimes y_i$ is an invariant identity tensor. Direct differentiation gives $C\iota=(n+1)I$, so the displayed <direct sum> follows. Its trace-free part is an <irreducible representation> of <highest weight> $\omega_1+2\omega_{n-1}$. Its <highest-weight vector> is $e_1\otimes y_n^2$; the <Weyl dimension formula> gives dimension $n(n-1)(n+2)/2$, exactly $\dim\ker C$.
// Target: semisimple-lie-algebra.bigb
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