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Abstract We consider finite-dimensional Hopf algebras $$u$$ that admit a smooth deformation $$U\to u$$ by a Noetherian Hopf algebra $$U$$ of finite global dimension. Examples of such Hopf algebras include small quantum groups over the complex numbers, restricted enveloping algebras in finite characteristic, and Drinfeld doubles of height $$1$$ group schemes. We provide a means of analyzing (cohomological) support for representations over such $$u$$, via the singularity categories of the hypersurfaces $U/(f)$ associated with functions $$f$$ on the corresponding parametrization space. We use this hypersurface approach to establish the tensor product property for cohomological support, for the following examples: functions on a finite group scheme, Drinfeld doubles of certain height 1 solvable finite group schemes, bosonized quantum complete intersections, and the small quantum Borel in type $$A$$.
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Ribbon operators in the generalized Kitaev quantum double model based on Hopf algebras
Abstract Kitaev’s quantum double model is a family of exactly solvable lattice models that realize two dimensional topological phases of matter. The model was originally based on finite groups, and was later generalized to semi-simple Hopf algebras. We rigorously define and study ribbon operators in the generalized quantum double model. These ribbon operators are important tools to understand quasi-particle excitations. It turns out that there are some subtleties in defining the operators in contrast to what one would naively think of. In particular, one has to distinguish two classes of ribbons which we call locally clockwise and locally counterclockwise ribbons. Moreover, we point out that the issue already exists in the original model based on finite non-abelian groups, but it seems to not have been noticed in the literature. We show how certain common properties would fail even in the original model if we were not to distinguish these two classes of ribbons. Perhaps not surprisingly, under the new definitions ribbon operators satisfy all properties that are expected. For instance, they create quasi-particle excitations only at the end of the ribbon, and the types of the quasi-particles correspond to irreducible representations of the Drinfeld double of the input Hopf algebra. However, the proofs of these properties are much more complicated than those in the case of finite groups. This is partly due to the complications in dealing with general Hopf algebras rather than group algebras.
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- Award ID(s):
- 2006667
- PAR ID:
- 10357262
- Date Published:
- Journal Name:
- Journal of Physics A: Mathematical and Theoretical
- Volume:
- 55
- Issue:
- 18
- ISSN:
- 1751-8113
- Page Range / eLocation ID:
- 185201
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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- Free Publicly Accessible Full Text
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Accepted Manuscript1.0
- Journal Article:
- https://doi.org/10.1088/1751-8121/ac552c
