= Graded symmetric algebra
{title2=$\operatorname{Sym}(V)$}
For a graded vector space $V$, $\operatorname{Sym}(V)$ is the quotient of the tensor algebra by $uv-(-1)^{|u||v|}vu$. Over $\mathbb Q$ it is polynomial on even-degree basis elements and exterior on odd-degree ones. With each generator primitive, it is a graded cocommutative Hopf algebra. Its primitive subspace is exactly $V$: applying the diagonal to a polynomial of symmetric length greater than one produces mixed terms, or equivalently the polynomial's formal derivatives cannot all vanish in characteristic zero. This is the associated graded algebra in the <graded Poincaré–Birkhoff–Witt theorem>.
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