If $\mathfrak{g}$ is a Lie algebra (over the field $k$ ), then we can construct its universal enveloping algebra $U(\mathfrak{g})$. We can define $\Delta:U(\mathfrak{g})\rightarrow U(\mathfrak{g})\otimes U(\mathfrak{g}) $ : $ x\mapsto 1\otimes x + x\otimes 1\in U(\mathfrak{g})\otimes U(\mathfrak{g})$. We can also define $\varepsilon:U(\mathfrak{g})\rightarrow k$ by $\varepsilon(x)=0, \mathfrak{g}\ni x\neq 1, \ \varepsilon(1)=1$.
Then $U\mathfrak(g)$ with its algebra and coalgebra structure can became a bialgebra,but bialgebra need we have $\Delta(xy)=\Delta(x)\Delta(y)$,that is to say $\Delta$ is a algebra homomorphism,which said that for $x,y\in U(\mathfrak{g})$,$1\otimes xy+xy\otimes 1=(1\otimes x+x\otimes 1)(1\otimes y+y\otimes 1)$,not a equality. Where is the mistake ?
Thankyou for sharing your mind.