# Two Hopf algebras associated to a linear algebraic group

Let $G$ be a linear algebraic group over a field $k$, then there are two different Hopf algebras associated to $G$. First is just coordinate ring $k[G]$of $G$. This is a commutative Hopf algebra. But also we can compute the Lie algebra $\mathfrak{g}$ and take the universal enveloping algebra $U(\mathfrak{g})$. This is a cocommutative Hopf algebra.

What is a relation between these two Hopf algebras? Is one dual to the other in some sense?

## 1 Answer

Let us assume that $\Bbbk$ is an algebraically closed field of characteristic $0$ and set $H:=\Bbbk[G]$. Consider the finite dual $H^\circ$ of $H$, i.e. $$H^\circ=\left\{f\in H^*\mid \ker(f)\supseteq I \textrm{ for a finite-codimensional ideal }I\subseteq H\right\}.$$ This is a cocommutative Hopf algebra. Since every cocommutative Hopf algebra over an algebraically closed field is pointed (all simple subcoalgebras are $1$-dimensional), it follows that $$H^\circ \cong U(P(H^\circ))\,\#\, \Bbbk G(H^\circ)$$ as Hopf algebras where $P(H^\circ)$ are the primitive elements of $H^\circ$, $G(H^\circ)$ are its group-like elements, $\Bbbk G(H^\circ)$ is the group algebra over $G(H^\circ)$ and $\#$ denotes the smash product. Up to isomorphism, one should then have that $P(H^\circ)=\mathfrak{g}$ and $G(H^\circ)=G$.

Therefore they are not exactly dual each other but they are still related.

As a reference for the above claims, you may have a look at Sweedler, Hopf algebras, section 13.1, and Abe, Hopf algebras, section 3.3 of Chapter 4.