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 [1] and [2] as references (the last of which is also given on the nlab) but [1] only really deals with ordinary Hopf algebras and [2] doesn’t seem to provide introductory material. I already found [3], which provides some background on dg-algebras, -coalgebras and -Lie algebras, but only little on dg-bialgebras and nothing on dg-Hopf algebras.
[1]: Loday, Cyclic Homology, Appendix A
[2]: Quillen, Rational Homotopy Theory, Appendix B
[3]: 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.