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

Question

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.

Answer 1

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.

Answer 2

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.