Kitaev’s Stabilizer Code and Chain Complex Theory of Bicommutative Hopf Algebras

The Formal Theory of Hopf Algebras

The category HopfC of Hopf monoids in a symmetric monoidal category C, assumed to be locally finitely presentable as a category, is analyzed with respect to its categorical properties. Assuming that the functors “tensor squaring” and “tensor cubing” on C preserve directed colimits one has the following results: (1) If, in C, extremal epimorphisms are stable under tensor squaring, then HopfC is ...

OB . K - Theory of Hopf Algebras

Homotopy Theory of Simplicial Abelian Hopf Algebras

We examine the homotopy theory of simplicial graded abelian Hopf algebras over a prime field Fp, p > 0, proving that two very different notions of weak equivalence yield the same homotopy category. We then prove a splitting result for the Postnikov tower of such simplicial Hopf algebras. As an application, we show how to recover the homotopy groups of a simplicial Hopf algebra from its André-Qu...

The Formal Theory of Hopf Algebras Part II: The case of Hopf algebras

The category HopfR of Hopf algebras over a commutative unital ring R is analyzed with respect to its categorical properties. The main results are: (1) For every ring R the category HopfR is locally presentable, it is coreflective in the category of bialgebras over R, over every R-algebra there exists a cofree Hopf algebra. (2) If, in addition, R is absoluty flat, then HopfR is reflective in the...

The Deformation Complex for DG Hopf Algebras

Let H be a DG Hopf algebra over a field k. This paper gives an explicit construction of a triple cochain complex that defines the HochschildCartier cohomology of H. A certain truncation of this complex is the appropriate setting for deforming H as an H (q)-structure. The direct limit of all such truncations is the appropriate setting for deforming H as a strongly homotopy associative structure....