Example 2

Let $\mathfrak{g}$ be a Lie algebra over a commutative field K and $U(\mathfrak{g})$ its enveloping algebra; we identify $\mathfrak{g}$ with its canonical image in $U(\mathfrak{g}) ($Lie, I, §2, No. 7, p. 39, Corollary 2). We endow $U(\mathfrak{g})$ with the structure of a bigebra with coproduct $c$ and counit $\gamma$ given by

for $\xi \in \mathfrak{g}($Lie, II, §1, No. 4, p. 115).

Let E and F be $U(\mathfrak{g}$)-modules; by formula (17), the $U(\mathfrak{g}$)-module structure on $E\otimes F$ is characterized by

for $\xi \in \mathfrak{g},x\in E$, and $y\in F$.

The Grothendieck ring $R_K(U(\mathfrak{g}))$ is also denoted by $\mathscr{R}(\mathfrak{g})$ in Lie, VIII,

§7, No. 6, p. $139.*$

Let A be a commutative ring. We can view A as a bigebra that is cocommutative over itself, with coproduct the natural isomorphism from A to $A\otimes_AA$ and counit Id$_A$ (III, §11, No. 4, p. 585). By Proposition 9, we obtain the following result.