# Why are there two Hopf algebra structures on a Kac--Moody Algebra.

For a Lie algebra $\frak{g}$, we will recall that its universal enveloping algebra $U(\frak{g})$ has a Hopf algebra structure given by $$\Delta(x) := 1 \otimes x + x \otimes 1, ~~~ \epsilon(x) = 0, ~~ S(x) = -x.$$

Now if $\frak{g}$ is a compact semi-simple Lie algebra, then its enveloping has another Hopf algebra structure, which deforms to give the famous Drinfeld--Jimbo algebras (see here for example).

What I would to know is: Does there exist any relationship between these two Hopf algebra structures? Moreover, does there exist some deformation of the first Hopf algebra to a noncommutative quantum group?

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These are the same Hopf algebra structure, written in terms of different generators. The $k$'s in the quantum group are exponentially of elements of the Lie algebra. –  Ben Webster Feb 16 '13 at 14:55
How does one take the exponent of the element of a Lie algebra? How is this well-defined without some type of topology? –  Dyke Acland Feb 16 '13 at 15:03
Sorry about the typo (stupid auto-complete). Anyways, this is too complicated a story to explain in a comment; the elements $k$ behave like they are $q^{H}$ for $H$ in the Lie algebra. There are various answers to what the hell that really means, but you can just calculate without worrying, and everything will be fine. –  Ben Webster Feb 16 '13 at 16:20
What's a good reference for this? –  Dyke Acland Feb 16 '13 at 16:41
Chari-Pressley's text is good. –  David Hill Mar 11 '13 at 13:14