Bicommutative Hopf algebras have internal hom objects. What are they? - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-24T00:30:45Z http://mathoverflow.net/feeds/question/56197 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/56197/bicommutative-hopf-algebras-have-internal-hom-objects-what-are-they Bicommutative Hopf algebras have internal hom objects. What are they? Tilman 2011-02-21T18:57:45Z 2011-02-22T14:03:54Z <p>Let $k$ be a field and $\mathcal H$ the category of conilpotent bicommutative Hopf algebras over $k$. (Conilpotent means that every element eventually becomes zero if you apply the reduced comultiplication $\tilde H \to \tilde H \otimes_k \tilde H$ repeatedly, $\tilde H$ being the augmentation ideal of $H$). $\mathcal H$ is an abelian category, and Goerss has constructed a tensor product in "Hopf rings, Dieudonné modules, and $E_*\Omega^2S^3$", Contemp. Math. 239, that represents bilinear maps of such Hopf algebras. He also notes that because of the special adjoint functor theorem, the tensor product has a right adjoint, an internal hom object thus.</p> <p>What is an explicit description of this internal hom? It seems to me that since $k[\mathbb Z]$ is the unit for the tensor product, and $Hom_{\mathcal H}(k[\mathbb Z],H)$ is the set of grouplike elements of $H$, the internal hom has to be some Hopf algebra whose grouplikes are precisely the Hopf algebra maps.</p> <p><em>Edit</em>: $k[\mathbb Z]$ is not actually conilpotent. So the category we should consider (and Goerss in fact considers) is nonnegatively graded Hopf algebras which are group rings in degree $0$.</p>