Graded symmetric algebra ($\mathrm{Sym}(V)$)

For a graded vector space $V$, $\mathrm{Sym}(V)$ 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.