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1. Braided Frobenius algebras from certain Hopf algebras

   https://doi.org/10.1142/S0219498823500123  

   Saito, Masahico ; Zappala, Emanuele ( September 2021 , Journal of Algebra and Its Applications)

   A braided Frobenius algebra is a Frobenius algebra with a Yang–Baxter operator that commutes with the operations, that are related to diagrams of compact surfaces with boundary expressed as ribbon graphs. A heap is a ternary operation exemplified by a group with the operation [Formula: see text], that is ternary self-distributive. Hopf algebras can be endowed with the algebra version of the heap operation. Using this, we construct braided Frobenius algebras from a class of certain Hopf algebras that admit integrals and cointegrals. For these Hopf algebras we show that the heap operation induces a Yang–Baxter operator on the tensor product, which satisfies the required compatibility conditions. Diagrammatic methods are employed for proving commutativity between Yang–Baxter operators and Frobenius operations. 

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2. Hopf–Hecke algebras, infinitesimal Cherednik algebras, and Dirac cohomology

   https://doi.org/10.4310/PAMQ.2021.v17.n4.a9  

   Flake, Johannes ; Sahi, Siddhartha ( January 2021 , Pure and Applied Mathematics Quarterly)
3. Degree bounds for Hopf actions on Artin–Schelter regular algebras

   https://doi.org/10.1016/j.aim.2022.108197  

   Kirkman, E. ; Won, R. ; Zhang, J.J. ( March 2022 , Advances in Mathematics)
4. The Jacobian, Reflection Arrangement and Discriminant for Reflection Hopf Algebras

   https://doi.org/10.1093/imrn/rnz380  

   Kirkman, E ; Zhang, J J ( January 2020 , International Mathematics Research Notices)

   Abstract We study finite-dimensional semisimple Hopf algebra actions on noetherian connected graded Artin–Schelter regular algebras and introduce definitions of the Jacobian, the reflection arrangement, and the discriminant in a noncommutative setting. 

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5. Ribbon operators in the generalized Kitaev quantum double model based on Hopf algebras

   https://doi.org/10.1088/1751-8121/ac552c  

   Yan, Bowen ; Chen, Penghua ; Cui, Shawn X ( April 2022 , Journal of Physics A: Mathematical and Theoretical)

   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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