Let $\mathfrak g$ be a Lie algebra. Show that on $U(\mathfrak g)$ (universal enveloping algebra) there is a natural Hopf algebra structure induced by the Hopf algebra structure on the tensor algebra $T(\mathfrak g)$.
So I know what the Hopf algebra structure on $T(\mathfrak g)$ is (the delta and epsilon functions, the antipode map, etc.). I can see how these maps induce maps on $U(\mathfrak g)$ since $U(\mathfrak g)$ is just a quotient on T(\mathfrak g), but my professor said we have to show they're well-defined. This is where I get stuck. Take the multiplication by tensors for example.
How would you show this is well-defined?
The enveloping algebra is a quotient of $T(\mathfrak{g})$ by the 2-sided ideal generated by $[x,y]-x\otimes y+y\otimes x$, $x,y\in\mathfrak{g}$. Since $\Delta$ is an algebra homomorphism, it is enough to show that $\Delta([x,y]-x\otimes y+y\otimes x)$ belongs to that ideal. Equivalently, you could show that $\Delta([x,y]) = \Delta(x\otimes y-y\otimes x)$ in $U(\mathfrak{g})$.
Well, since $[x,y]\in\mathfrak{g}$, $\Delta([x,y])=[x,y]\otimes 1 + 1\otimes [x,y]$. The harder computation is $$ \Delta(x\otimes y-y\otimes x)=\Delta(x)\otimes \Delta(y)-\Delta(y)\otimes \Delta(x). $$ Write it out and use the relations in $U(\mathfrak{g})$ (namely, $\cdots\otimes(x\otimes y-y\otimes x)\otimes\cdots=\cdots\otimes[x,y]\otimes\cdots$) to show that it equals $\Delta([x,y])$ as written above.