Different Hopf algebras I have a stupid question, but for a Hopf algebra $A$: $A$ and $A\otimes A$ are different Hopf algebras right? As in, being a Hopf algebra simply equips some maps from $A$ and gives us some rules (i.e. three diagrams each for unit, product, counit, coproduct, and extra diagrams for antipode and compatibility). But being a Hopf algebra says little about the algebras $A\otimes^k A$, although these can be given the structure of different Hopf algebras?
I got this confusion from thinking about the universal enveloping algebra, which being $T(\mathfrak{g})/(xy-yx-[x,y])$, say we have $\Delta(x)=x\otimes x$ then it maps from $U(\mathfrak{g})\to U(\mathfrak{g})\otimes U(\mathfrak{g})$ but it is also in $U(\mathfrak{g})$ right?

If I got the point, your problem comes from the fact that the product in $T(\mathfrak{g})$ is given by concatenation, so for $x,y\in\mathfrak{g}$ you may look at $x\otimes y$ both as an element in $T^1(\mathfrak{g})\otimes T^1(\mathfrak{g})$ and in $T^2(\mathfrak{g})$. Formally, you should distinguish the concatenation product $x\cdot y=x\otimes y \in T^2(\mathfrak{g})$ from the "external tensor product" $x\otimes y \in T^1(\mathfrak{g})\otimes T^1(\mathfrak{g})$. What you can say is that the vector spaces $T^1(\mathfrak{g})\otimes T^1(\mathfrak{g})$ and $T^2(\mathfrak{g})$ are isomorphic (exactly via the multiplication) and sometimes you may identify them.
As an example of the importance of distinguish the two tensor products, you may look at the tensor product of two linear maps $f:V\to V'$ and $g:W\to W'$. The writing $f\otimes g$ may mean both the linear map $f\otimes g:V\otimes W\to V'\otimes W'$ and the element $f\otimes g\in \mathsf{Hom}(V,V')\otimes \mathsf{Hom}(W,W')$.
