I have two questions about the definition of a Hopf algebra in Hatcher's book on algebraic topology. He defines it as follows (see Section 3.C, page 283):

Definition: A Hopf algebra is a graded algebra $A = \oplus_{n \geq 0} A^n$ over a commutative base ring $R$ satisfying the following two conditions:

  1. There is an identity element $1 \in A^0$ such that the map $R \rightarrow A^0$, $r \mapsto r \cdot 1$, is an isomorphism; one says that $A$ is connected.
  2. There is a diagonal or coproduct $\Delta\colon A \rightarrow A \otimes A$, a homomorphism of graded algebras satisfying $\Delta(\alpha) = \alpha \otimes 1 + 1 \otimes \alpha + \sum_{0<i<n} \alpha'_i \otimes \alpha''_{n-i}$ for $\alpha \in A^n$, $n >0$, and $\alpha'_j,\alpha''_j \in A^j$.

Of course, in other settings Hopf algebras don't need to be graded, but I guess graded ones are the ones he cares about. Here are my two questions:

  1. In condition 2, one could instead ask for the a priori weaker condition that $\Delta(\alpha) = \alpha \otimes 1 + 1 \otimes \alpha + x$ with $x \in \oplus_{0<i<n} A^i \otimes A^{n-i}$. It seems to me that in his examples (all coming from H-spaces) this is all that holds, at least naively. Is this a mistake, or is this really the right definition?

  2. When I look at definitions of Hopf algebras elsewhere, there is also an antipode map. How can you derive this in Hatcher's setup?

  • 8
    $\begingroup$ Regarding 1: You are right about the notation being incorrect. $\alpha_i \otimes \alpha_{n-i}$ is really a placeholder for a linear combination of elements of this form. Regarding 2: It is a standard exercise, and worth doing, that in this connected graded setting, the existence of the antipode can be deduced. (One figures out the formula for the antipode by induction on the degree of an element) $\endgroup$ – Nicholas Kuhn Mar 4 at 17:06

You have not transcribed Hatcher's definition correctly. He has a sum $\sum_i\alpha'_i\otimes\alpha''_i$ over some unspecified index set, not over $\{1,\dotsc,n-1\}$, and he requires that the terms $\alpha'_i,\alpha''_i$ are homogeneous of arbitrary positive degree (although the requirement that $\Delta$ preserves gradings forces $|\alpha'_i|+|\alpha''_i|=|\alpha|$. With these corrections, Hatcher's definition is equivalent to your "a priori weaker condition".

Also, in the connected graded context, it is not hard to check by induction on $n$ that there is a unique way to define the antipode $\chi\colon A^n\to A^n$ such that the defining property $\mu\circ(\chi\otimes 1)\circ\Delta=\eta\epsilon$ is satisfied.

  • 17
    $\begingroup$ I transcribed the definition word-for-word from my printed copy of his book. But looking at his website, the version has what you say. His errata (pi.math.cornell.edu/~hatcher/AT/AT-errata.pdf) clears this up: this was a genuine mistake in his printed version, and is corrected in the online one. $\endgroup$ – Irina Mar 4 at 17:15

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