= 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 a <Hopf algebra> $H$ is primitive when
$$
\Delta x=x\otimes1+1\otimes x.
$$
No grading is required. Applying the <counit> to either tensor factor shows $\varepsilon(x)=0$. If $x,y$ are primitive, the algebra-homomorphism property of <comultiplication> gives
$$
\Delta[x,y]=[x,y]\otimes1+1\otimes[x,y],
$$
because elements in the two separate tensor factors commute. Thus the primitive elements form a <Lie algebra> under the <commutator>.
In a <universal enveloping algebra> over $\mathbb Q$, the primitive elements are exactly the original <Lie algebra>. The <Poincaré-Birkhoff-Witt theorem> identifies the associated graded algebra with a <symmetric algebra>; in characteristic zero, a nonzero homogeneous <polynomial> of degree greater than one cannot be additive under the diagonal. The highest-degree part of a primitive element must therefore have degree one. Positive <characteristic of a field> changes this conclusion: if $x$ is primitive in characteristic $p$, then $x^p$ is primitive too, since the intermediate <binomial coefficients> vanish modulo $p$.
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