Primitive element of a Hopf algebra ($\Delta x=x\otimes 1+1\otimes x$)

An element $x$ of a Hopf algebra $H$ is primitive whenNo grading is required. Applying the counit to either tensor factor shows $\epsilon (x)=0$. If $x,y$ are primitive, the algebra-homomorphism property of comultiplication givesbecause 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 $p$.