Graded commutative algebra

A graded algebra is a graded commutative algebra if $uv=(-1{)}^{pq}vu$ for homogeneous elements of a graded algebra of degrees $p,q$. In characteristic not equal to two, every odd-degree element then squares to zero. The exterior algebra and Hochschild cohomology with the Hochschild cup product are examples.

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.