Hopf Algebra

A hopf algebra $A$ is a graded algebra $A=\bigoplus_{n\geq 0}A_{n}$ over a commutative ring $R$ so that there exists an identity $1\in A_{0}$ such that the map $R\to A_{0}$ defined by $r\mapsto r\cdot 1$ is an isomorphism and there exists a homomorphism of graded algebra $\Delta:A\to A\otimes A$ so that

$\Delta (\alpha)=\alpha\otimes 1+1\otimes \alpha+\sum_{0<i<n}\alpha_{i}'\otimes \alpha_{n-i}''$

for $\alpha\in A_{n}$ , $n>0$ and $\alpha_{j}',\alpha_{j}''\in A_{j}$ .

Let $A=R[\alpha]$ be the polynomial ring over $R$ . Then $\Delta(\alpha)=\alpha\otimes 1+1\otimes \alpha$ . Assume that $\alpha$ is odd dimensional. Then $(\alpha \otimes 1)(1\otimes \alpha)=\alpha\otimes\alpha$ and $(1\otimes\alpha)(\alpha\otimes 1)=-\alpha\otimes\alpha$ . This would implies that $\Delta(\alpha)^{2}=\alpha^{2}\otimes 1+1\otimes\alpha^{2}=\Delta(\alpha^{2})$ .

Let $\Lambda_{R}[\alpha]$ be the exterior algebra over $R$ . Then $\alpha^{2}=0$ . It is easy to see that $\Delta(\alpha)^{2}=0$ .

An element $\alpha$ in a Hopf algebra $A$ is said to be primitive if $\Delta\alpha=\alpha\otimes 1+1\otimes\alpha$ .

Exercise: Let $A,B$ be hopf algebras over $R$ . On $A\otimes_{R} B$ , we define $\Delta(a\otimes b)=\Delta(a)\otimes\Delta(b)$ . Show that $A\otimes_{R}B$ is again a hopf algebra.