# Is a comultiplication structure unique?

Let $$A$$ be an $$R$$-algebra.

Suppose $$A$$ has a $$R$$-coalgebra structure compatible with the algebra structure. (I.e. there is a comultiplication map $$\Delta$$ and counit map $$\epsilon$$ compatible with the multiplication map and unit map of $$A$$.)

Then I am wondering whether $$A$$ can have another $$R$$-coalgebra structure other than $$(\Delta, \epsilon)$$?

Yes, $$A$$ can have multiple compatible $$R$$-coalgebra structures. Let $$G$$ be a finite set and let $$A=R(G)$$ be the commutative algebra of functions on $$G$$ with pointwise multiplication. Then any group structure on $$G$$ gives an Hopf structure on $$A$$ and non-isomorphic group structures on $$G$$ leads to non-isomorphic Hopf structures on $$A$$.
• Is it true for all $R$ algebra not only of the form $R(G)$? I am especially considering the universal enveloping algebra. Is it true for universal enveloping algebra? – Monty Apr 16 at 16:12
• Yes, e.g. take the trivial Lie algebra, then you just get a polynomial algebra and those have many Hopf structures. If you start with a simple Lie algebra $\mathfrak g$ and let $R=\mathbb{C}[\epsilon]/(\epsilon^2)$ then the standard Lie bialgebra structure on $\mathfrak g$ gives you another example. I'm sure there are plenty more. – Adrien Apr 16 at 16:24
• Sorry, I am not sure whether I understood well your comment. You say that there might be some Lie algebra $\mathfrak{g}$ such that $U(\mathfrak{g})$ may have many Hopf algebra structure? Right? I guessed that $U(\mathfrak{g})$ has only one Hopf algebra structure! – Monty Apr 16 at 16:33
• yes that's what I'm saying, and there are examples already when $\mathfrak g$ is abelian, i.e. have the trivial Lie structure. – Adrien Apr 16 at 16:42
• If you want specifics : let $\mathfrak g=R$ be the one dimensional Lie algebra with zero bracket. Then $U(\mathfrak g)$ is just $R[x]$ as an algebra, and this has a Hopf structure given by $\Delta(x)=x \otimes x$ which is not the one coming from being an enveloping algebra. – Adrien Apr 16 at 16:44