arXiv subjects
Publications and source records attributed to Ulrich Krähmer.
This article considers cuspidal curves whose coordinate rings are numerical semigroup algebras. Using a general result about descent of Hopf algebroid structures, their rings of differential operators are shown to be cocommutative and conilpotent left Hopf algebroids. If the semigroups are symmetric so that the curves are Gorenstein, they are full Hopf algebroids (admit an antipode).
The construction of the topologically protected code space of Kitaev's model for fault-tolerant quantum computation is extended from complex semisimple to arbitrary finite-dimensional Hopf algebras admitting pairs in involution. One input of the model are ribbon graphs, that is, the combinatorial data of cellular decompositions of oriented closed surfaces. The other input are certain Hopf bimodules that are closely related to the coefficients in Hopf-cyclic homology. As in previous generalisations of the Kitaev model, a Yetter-Drinfeld module is constructed and shown to be a topological invariant of the surface with boundary that is obtained by "thickening" the graph. The generalisation of the protected space is defined using bitensor products of modules-comodules. Provided that the Hopf bimodule coefficients correspond to pairs in involution, this is shown to depend only on the genus of the graph. As examples, group algebras of finite groups and bosonisations of Nichols algebras are considered.
Böhm and Ştefan have expressed cyclic homology as an invariant that assigns homology groups $\mathrm{HC}^χ_i(\mathrm N, \mathrm M)$ to right and left coalgebras $\mathrm N$ respectively $\mathrm M$ over a distributive law $χ$ between two comonads. For the key example associated to a bialgebra $H$, right $χ$-coalgebras have a description in terms of modules and comodules over $H$. The present article formulates conditions under which such a description is simultaneously possible for the left $χ$-coalgebras. In the above example, this is the case when the bialgebra $H$ is a Hopf algebra with bijective antipode. We also discuss how the generalized Hopf module theorem by Mesablishvili and Wisbauer features both in theory and examples.
The fact that the cocommutative comonoids in a symmetric monoidal category form the best possible approximation by a cartesian category is revisited when the original category is only braided monoidal. This leads to the question when the endomorphism operad of a comonoid is a clone (a Lawvere theory). By giving an explicit example, we prove that this does not imply that the comonoid is cocommutative.
Let $A$ be a Hopf algebra equipped with a projection onto the coordinate Hopf algebra $\mathcal{O}(G)$ of a semisimple algebraic group $G$. It is shown that if $A$ admits a suitably non-degenerate comodule $V$ and the induced $G$-module structure of $V$ is non-trivial, then the third Hochschild homology group of $A$ is non-trivial.
This article studies the construction of Hopf algebras $H$ acting on a given algebra $K$ in terms of algebra morphisms $ σ\colon K \rightarrow \mathrm{M}_n(K)$. The approach is particularly suited for controlling whether these actions restrict to a given subalgebra $B$ of $K$, whether $H$ is pointed, and whether these actions are compatible with a given $*$-structure on $K$. The theory is applied to the field $K=k(t)$ of rational functions containing the coordinate ring $B=k[t^2,t^3]$ of the cusp. An explicit example is described in detail and shown to define a quantum homogeneous space structure on the cusp, which, unlike the previously known one, extends from regular to rational functions.
The self-duality of the paracyclic category is extended to a certain class of homotopy categories of (2,1)-categories. These generalise the orbit category of a group and are associated to certain self-dual preorders equipped with a presheaf of groups and a cosieve. Slice 2-categories of equidimensional submanifolds of a compact manifold without boundary form a particular case, and for $S^1$, one recovers cyclic duality. This provides in particular a visualisation of the results of Böhm and Ştefan on the topic.