# Reference Request: Differential Graded Hopf Algebras

To prepare an upcoming seminar talk I’m trying to find introductory texts on differential graded Hopf algebras (over a field $$k$$).

My knowledge of Hopf algebras encompasses roughly the first 5 chapters of Sweedler’s book. I also have a rough idea of how to define dg-Hopf algebras, namely by the same diagrams as an ordinary Hopf algebra but with dg-vector spaces instead of just vector spaces (i.e. as “Hopf objects” in the monoidal category of dg-vector spaces). I’m thus looking for references that do the following:

• Providing an explicit definition of dg-Hopf algebras and showing their fundamental properties.
• Giving (basic) examples for dg-Hopf algebras.
• Explaining the differences and similarities between ordinary Hopf-algebras and dg-Hopf algebras.
• (Giving me a way to check my thoughts and suspicions.)

I was originally given  and  as references (the last of which is also given on the nlab) but  only really deals with ordinary Hopf algebras and  doesn’t seem to provide introductory material. I already found , which provides some background on dg-algebras, -coalgebras and -Lie algebras, but only little on dg-bialgebras and nothing on dg-Hopf algebras.

: Loday, Cyclic Homology, Appendix A
: Quillen, Rational Homotopy Theory, Appendix B
: Félix, Halperin, Thomas, Rational Homotopy Theory, Chapters 3,21

I cannot give you a reference better than "wikipedia definition of Hopf algebra in a braided category" and it specialization to the braiding $$v\otimes w\mapsto (-1)^{|v||w|}w\otimes v$$. But I can give you the first elementary and nontrivial example of d.g. Hopf algebra that is not a Hopf algebra in the usual sense:
Fix a field of characteristic not 2 and $$H=k[x]/x^2$$. Then $$H$$ is not a Hopf algebra, because it is commutative and has nontrivial nilpotent elements. On the other hand, it is a d.g. Hopf algebra with $$|1|=0$$, $$|x|=1$$ and $$d(x)=1$$, simply defining $$\Delta(x)=x\otimes 1 +1\otimes x$$
The reason is that, if you define $$\Delta$$ as above, the way you extend it to $$k[x]$$ in the d.g. setting is considering the algebra structure in $$k[x]\otimes k[x]$$ given by $$(x^n\otimes x^m)\cdot (x^i\otimes x^j)=(-1)^{im}x^{n+1}\otimes x^{n+j}$$ With this structure, the element $$x^2$$ mapsto $$(x\otimes 1+1\otimes x) ^2 =x^2\otimes 1+x\otimes x -x\otimes x+1\otimes x^2=x^2\otimes 1+1\otimes x^2$$ (and not to $$x^2\otimes 1+2x\otimes x+1\otimes x^2$$) So, the "super sign" makes the job so that the ideal $$(x^2)$$ is also a coideal, and hence $$\Delta:k[x]/x^2\to k[x]/x^2\otimes k[x]/x^2$$ is a well-defined algebra map, when considering $$k[x]/x^2\otimes k[x]/x^2$$ as algebra using the Koszul signs.
• I don’t see how $d(x) = 1$ is supposed to work: Then $1 \in k[x]/x^2$ is a boundary but $1 \in k$ is not, so the counit $\varepsilon \colon k[x]/x^2 \to k$ cannot be a morphism of dg-algebras. I think it should be $d(x) = 0$, so that we get the dg-symmetric algebra on a one-dimensional dg-vector space concentrated in degree 1 (if I understand this correctly). – Jendrik Stelzner May 18 at 10:10
You are right. $$\epsilon\circ d=0$$ for any coderivation d, in the same way as d(1)=0 for a derivation d. Fortunately, with dx=0 the example survives