A character of an algebra is a nonzero linear functional which is also multiplicative, that is,

$$\mu(ab)=\mu(a)\,\mu(b)\quad\text{for all}\quad a,b;$$

notice that $\mu(1)=1$ . The counit $\varepsilon$ of a bialgebra is a character. Characters of a bialgebra can be convolved, since $\mu*\nu=(\mu\otimes\nu)\Delta$ is a composition of homomorphisms. The characters of a Hopf algebra $H$ form a group $\mathcal{G}(H)$ under convolution, whose neutral element is $\varepsilon$ ; the inverse of $\mu$ is $\mu S$ .

A derivation or “infinitesimal character” of a Hopf algebra $H$ is a linear map $\delta:H\to\mathbb{F}$ satisfying

$$\delta(ab)=\delta(a)\varepsilon(b)+\varepsilon(a)\delta(b)\quad\text{for all}\quad a,b\in H.$$

This entails $\delta(1_{H})=0$ . The previous relation can also be written as $m^{t}(\delta)=\delta\otimes\varepsilon+\varepsilon\otimes\delta$ , which shows that $\delta$ belongs to $H^{\circ}$ and is primitive there; in particular, the bracket $[\delta,\partial]:=\delta*\partial-\partial*\delta$ of two derivations is again a derivation. Thus the vector space $\operatorname{Der}_{\varepsilon}(H)$ of derivations is actually a Lie algebra.