Hopf-algebra structure on U(g)

The assignments on $x\in \mathfrak{g}$

$\mathrm{\Delta }(x)={\iota }_{\mathfrak{g}}(x)\otimes 1+1\otimes {\iota }_{\mathfrak{g}}(x),\epsilon (x)=0,S(x)=-{\iota }_{\mathfrak{g}}(x)$

extend to the standard cocommutative Hopf-algebra structure on $U(\mathfrak{g})$. The first two assignments are Lie maps into the commutator algebras of $U(\mathfrak{g})\otimes U(\mathfrak{g})$ and $k$, so enveloping universality extends them to algebra maps. Interpreting the last assignment as a Lie map into $U(\mathfrak{g}{)}^{\mathrm{o}\mathrm{p}}$ extends it to an algebra map into the opposite algebra, equivalently an anti-algebra map $S$.

Coassociativity, cocommutativity, and the counit identities follow because the corresponding algebra maps agree on every generator. For the antipode, write $\mathrm{\Delta }(a)=\sum a_{(1)}\otimes a_{(2)}$. Both convolution identities hold on $1$ and on every generator, since multiplying $-x\otimes 1+1\otimes x$ gives zero. If they hold for $a$ and $b$, then

$\sum S(a_{(1)}{b}_{(1)})a_{(2)}{b}_{(2)}=\sum S(b_{(1)})S(a_{(1)})a_{(2)}{b}_{(2)}=\epsilon (a)\epsilon (b)1,$

and similarly

$\sum a_{(1)}{b}_{(1)}S(a_{(2)}{b}_{(2)})=\sum a_{(1)}{b}_{(1)}S(b_{(2)})S(a_{(2)})=\epsilon (a)\epsilon (b)1.$

Induction on word length and linearity therefore prove both antipode identities on all of $U(\mathfrak{g})$.

This remark is non-load-bearing: no later item on this page depends on these Hopf formulas.