= Primitive element of a Hopf algebra
{title2=$\Delta x=x\otimes1+1\otimes x$}
= Primitive elements of a Hopf algebra
{synonym}
An element $x$ of positive degree is primitive when $\Delta x=x\otimes1+1\otimes x$. Graded <commutators> of primitives are primitive, so they form a <graded Lie algebra>. In a <universal enveloping algebra> over $\mathbb Q$, the <primitive elements of a Hopf algebra> are exactly its given <Lie algebra>: the associated graded algebra of the word-length filtration is symmetric, where no nonzero homogeneous polynomial of length greater than one is additive under the diagonal.
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