Antipode of a Hopf algebra satisfies $\sum_{(x)} S(x') \otimes x'' \otimes x''' = \sum_{(x)} \epsilon(x') \otimes x''$

Question

Page 51 of Christian Kassel’s “Quantum Groups” contains the following:

Let $(H, \mu, \nu, \Delta, \epsilon, S)$ be a Hopf Algebra. Then for all $x$ in $H$ we have $\sum_{(x)} S(x') x'' = \epsilon(x) 1$ where $\epsilon$ is the counit. Using Sweedlers notation, it follows that $$ \sum_{(x)} S(x') \otimes x'' \otimes x''' = \sum_{(x)} \epsilon(x') \otimes x'' \,. $$

Can somebody fill in the gaps for me here? To me it seems like we would need $\sum_{(x)} S(x') \otimes x'' = \epsilon(x) 1$ to be true in order to draw the given conclusion, but all we have is $\sum_{(x)} S(x') x'' = \epsilon(x) 1$. Thanks!

Answer 1

As already pointed out in the comments, this is an error in the book. The errata at https://irma.math.unistra.fr/~kassel/QGbk.html state the following:¹

• page 51, line 12–13: these two lines should be \begin{align*} \sum_{(x)} x^{(1)} ⊗ x^{(2)} S(x^{(3)}) ⊗ x^{(4)} ⊗ x^{(5)} &= \sum_{(x)} x^{(1)} ⊗ ε(x^{(2)}) 1 ⊗ x^{(3)} ⊗ x^{(4)} \\[0.3em] &= \sum_{(x)} x^{(1)} ⊗ 1 ⊗ x^{(2)} ⊗ x^{(3)} \,. \end{align*}

These equations are correct because \begin{align*} {}& \sum_{(x)} x^{(1)} ⊗ x^{(2)} S(x^{(3)}) ⊗ x^{(4)} ⊗ x^{(5)} \tag{first term} \\[0.3em] ={}& \sum_{(x)} \sum_{(x^{(2)})} x^{(1)} ⊗ x^{(2)(1)} S(x^{(2)(2)}) ⊗ x^{(3)} ⊗ x^{(4)} \\ ={}& \sum_{(x)} x^{(1)}⊗ \Biggl( \sum_{(x^{(2)})} x^{(2)(1)} S( x^{(2)(2)} ) \Biggr) ⊗ x^{(3)} \\[0.3em] ={}& \sum_{(x)} x^{(1)} ⊗ ε(x^{(2)}) 1 ⊗ x^{(3)} ⊗ x^{(4)} \tag{second term} \\[0.3em] ={}& \sum_{(x)} x^{(1)} ε(x^{(2)}) ⊗ 1 ⊗ x^{(3)} ⊗ x^{(4)} \\ ={}& \sum_{(x)} \sum_{(x^{(1)})} x^{(1)(1)} ε(x^{(1)(2)}) ⊗ 1 ⊗ x^{(3)} ⊗ x^{(4)} \\ ={}& \sum_{(x)} \Biggl( \sum_{(x^{(1)})} x^{(1)(1)} ε(x^{(1)(2)}) \Biggr) ⊗ 1 ⊗ x^{(3)} ⊗ x^{(4)} \\ ={}& \sum_{(x)} x^{(1)} ⊗ 1 ⊗ x^{(2)} ⊗ x^{(3)} \,. \tag{third term} \end{align*} (I use the abbreviation $x^{(i)(j)}$ for $( x^{(i)} )^{(j)}$.)

---

Out of curiosity, I also checked the copy of Kassel’s Quantum Groups that I have access to. It states the following:

Using Sweedler’s convention 1.6, we see that an antipode satisfies the relation $$ \sum_{(x)} x' S(x'') = ε(x) 1 = \sum_{(x)} S(x') x'' \tag{3.3} $$ for all $x ∈ H$. In any Hopf algebra we have relations such as \begin{align*} \sum_{(x)} x^{(1)} ⊗ x^{(2)} ⊗ S(x^{(3)}) ⊗ x^{(4)} ⊗ x^{(5)} &= \sum_{(x)} x^{(1)} ⊗ ε(x^{(2)}) ⊗ x^{(3)} ⊗ x^{(4)} \\[0.3em] &= \sum_{(x)} x^{(1)} ⊗ x^{(2)} ⊗ x^{(3)} \,. \end{align*}

These equations are still not correct (as can be seen from the inconsistent number of tensor symbols), but already closer to the correct equations from the errata. (I guess these still-incorrect equations are what Elliot Yu eludes to in the comments: “It seems like the original relation in the book is missing a $μ$ on the left and has an extra $μ$ on the right.”)

---

¹ At the time of writing, the PDF version of the errata can’t be accessed anymore, but the PS version is still available.