Related papers: Coribbon Hopf (face) algebras generated by lattice…
The modular group algebra of an elementary abelian p-group is isomorphic to the restricted enveloping algebra of commutative restricted Lie algebra. The different ways of regarding this algebra result in different Hopf algebra structures…
In this paper we describe the right-sided combinatorial Hopf structure of three Hopf algebras appearing in the context of renormalization in quantum field theory: the non-commutative version of the Fa\`a di Bruno Hopf algebra, the…
Any finite-dimensional Hopf algebra H is Frobenius and the stable category of H-modules is triangulated monoidal. To H-comodule algebras we assign triangulated module-categories over the stable category of H-modules. These module-categories…
This is a survey on spherical Hopf algebras. We give criteria to decide when a Hopf algebra is spherical and collect examples. We discuss tilting modules as a mean to obtain a fusion subcategory of the non-degenerate quotient of the…
Given a Hopf algebra $A$ graded by a discrete group together with an action of the same group preserving the grading, we define a new Hopf algebra, which we call the graded twisting of $A$. If the action is adjoint, this new Hopf algebra is…
Quantum contextuality, a fundamental feature distinguishing quantum theory from classical models, is investigated via algebraic and topological structures inherent in modular tensor categories. This work rigorously demonstrates that braid…
We describe certain quiver Hopf algebras by parameters. This leads to the classification of multiple Taft algebras as well as pointed Yetter-Drinfeld modules and their corresponding Nichols algebras. In particular, when the ground-field $k$…
We classify the cosemisimple Hopf algebras whose corepresentation semi-ring is isomorphic to that of GL(2). This leads us to define a new family of Hopf algebras which generalize the quantum similitude group of a non-degenerate bilinear…
The ribbon cocycle invariant is defined by means of a partition function using ternary cohomology of self-distributive structures (TSD) and colorings of ribbon diagrams of a framed link, following the same paradigm introduced by Carter,…
We classify braided tensor categories over C of exponential growth which are quasisymmetric, i.e., the squared braiding is the identity on the product of any two simple objects. This generalizes the classification results of Deligne on…
We propose a nonperturbative construction of Hopf algebras that represent categories of line operators in topological quantum field theory, in terms of semi-extended operators (spark algebras) on pairs of transverse topological boundary…
An important property of a Hopf algebra is its quasitriangularity and it is useful various applications. This property is investigated for quantum groups $sl_2$ at roots of 1. It is shown that different forms of the quantum group $sl_2$ at…
We classify Nichols algebras of irreducible Yetter-Drinfeld modules over groups such that the underlying rack is braided and the homogeneous component of degree three of the Nichols algebra satisfies a given inequality. This assumption…
This paper introduces methods for classifying actions of finite-dimensional Hopf algebras on path algebras of quivers, and more generally on tensor algebras $T_B(V)$ where $B$ is semisimple. We work within the broader framework of finite…
Sub-Riemannian structures on odd-dimensional spheres respecting the Hopf fibration naturally appear in quantum mechanics. We study the curvature maps for such a sub-Riemannian structure and express them using the Riemannian curvature tensor…
The paper presents applications of Toeplitz algebras in Noncommutative Geometry. As an example, a quantum Hopf fibration is given by gluing trivial U(1) bundles over quantum discs (or, synonymously, Toeplitz algebras) along their…
We discuss groups corresponding to Kohno Lie algebra of infinitesimal braids and actions of such groups. We construct homomorphisms of Lie braid groups to the group of symplectomorphisms of the space of point configurations in $R^3$ and to…
The Reshetikhin-Turaev invariant, Turaev's TQFT, and many related constructions rely on the encoding of certain tangles (n-string links, or ribbon n-handles) as n-forms on the coend of a ribbon category. We introduce the monoidal category…
This is a survey on the state-of-the-art of the classification of finite-dimensional complex Hopf algebras. This general question is addressed through the consideration of different classes of such Hopf algebras. Pointed Hopf algebras…
I report the recent advances in applying (graded) Hopf algebras with braided tensor product in two scenarios: i) paraparticles beyond bosons and fermions living in any space dimensions and transforming under the permutation group; ii)…