What is $\Delta(1)$ for $1$ in $U(\mathfrak{g})$?

Let $\mathfrak{g}$ be a semisimple Lie algebra and $U(\mathfrak{g})$ its universal enveloping algebra. Then $U(\mathfrak{g})$ is a hopf algebra. Is $\Delta(1) = 1 \otimes 1$ or $\Delta(1) = 1 \otimes 1 + 1 \otimes 1$? Here $1$ is the identity in $U(\mathfrak{g})$. Thank you very much.

• By definition of a Hopf algebra, the comultiplication map is an algebra homomorphism, so $\Delta(1)=1\otimes 1$. This is not to be confused with the comultiplication of Frobenius algebras, which are not algebra homomorphisms. – Hanno Oct 26 '14 at 14:27
• @Hanno, thank you very much. Yes, I think we have $\Delta(1) = 1 \otimes 1$ since $1 \otimes 1$ is the identity of $U(\mathfrak{g}) \otimes U(\mathfrak{g})$ and $\Delta$ is a homomorphism of algebras. – LJR Oct 30 '14 at 7:15

One of the axioms of a bialgebra $B$ (in particular a Hopf algebra) is:

(taken from Wikipedia: in their notation $K$ is the base field and $B$ is the bialgebra).

Here $\eta$ is the unit and $\Delta$ is coproduct. In plain words, $\eta(\lambda \in K) = \lambda 1_B$. By the commutativity of the diagram, $$\Delta(1_B) = \Delta(\eta(1_K)) = (\eta \otimes \eta)(1_K \otimes 1_K) = 1_B \otimes 1_B.$$

As Hanno says you can interpret that by remembering that $\Delta$ is a algebra homomorphism in a bialgebra, so it has to preserve the unit.

The equation $\Delta(x) = 1 \otimes x + x \otimes 1$ is only valid for the elements of $\mathfrak{g}$ in $U(\mathfrak{g})$. Such an element is called a primitive element. You should be very careful when computing the coproduct in $U(\mathfrak{g})$, because it's tempting to believe that since $\Delta(x) = 1 \otimes x + x \otimes 1$ on generators, the same holds true for every element. That's not the case. The rule to use is again that $\Delta$ is a algebra homomorphism, so for example if $x, y \in \mathfrak{g}$, $$\Delta(xy) = \Delta(x) \Delta(y) = (1 \otimes x + x \otimes 1) (1 \otimes y + y \otimes 1) = 1 \otimes xy + x \otimes y + y \otimes x + xy \otimes 1.$$ And a similar pattern holds true for products with more terms.

On my master thesis two year ago, i think i used $\Delta(1) = 1 \otimes 1 + 1 \otimes 1$. By a fast googling, i found the same on wiki.

I hope that helps.

• The equality $\Delta(X) = X\otimes 1 + 1\otimes X$ holds for $X\in {\mathfrak g}\subset{\mathscr U}{\mathfrak g}$, but not for the unit $1$ of ${\mathscr U}{\mathfrak g}$. As said in the comments, the unit has to be preserved under comultiplication, and the unit of ${\mathscr U}{\mathfrak g}\otimes{\mathscr U}{\mathfrak g}$ is $1\otimes 1$. – Hanno Oct 30 '14 at 8:00
• Hi Giannis. Nice to see you here! Btw: I am sure you have used the correct expression $\Delta(1) = 1 \otimes 1$ in your master thesis. (I verified this by looking at the bottom of p. 40 of your thesis as this is posted in the arXiv) ;) – KonKan Jan 7 '17 at 20:57