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A Hopf algebra $H$ (over field $k$) is a bialgebra $(H,m,u,\Delta,\epsilon)$ (H, product, unit, coproduct, counit), with an antipode map, $S:H\to H$ such that $$\sum (Sh_{(1)})h_{(2)} = \epsilon(h) = \sum h_{(1)}(Sh_{(2)}), \forall h\in H$$

This antipode map doesn't make sense to me: $\epsilon:H\to k$, and $S:H\to H$, so then the LHS and RHS are in $H$, and they equal something in $k$?

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What you have, in fact (see e.g. the definition on Wikipedia) is: $$\sum (Sh_{(1)}) h_{(2)} = u(\epsilon(h)) = \dots$$

You need to use the unit to then make an element of $H$ from an element of $k$. Perhaps your book is implicitly using the unit here, because the equation couldn't really mean anything else: the image of $u$ inside $H$ is canonically identified with $k$. But if you want to be completely rigorous you need to write down the unit.

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  • $\begingroup$ Thank you, indeed my text must be implicitly using it, and I wasn't sure what that $1$ was doing on wikipedia before I asked(I assumed it was the $1\in k$, which wouldn't have helped). $\endgroup$ – Katie Feb 3 '16 at 14:15

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