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Universal $T$-matrices, or Hopf algebra dual forms, for quantum groups are revisited, and their contraction theory is developed. As a first illustrative example, the (1+1) timelike $\kappa$-Poincar\'e $T$-matrix is explicitly worked out.…

We find a self-dual noncommutative and noncocommutative Hopf algebra acting as a universal symmetry on the modules over inner Frobenius algebras of modular categories (as used in two dimensional boundary conformal field theory) similar to…

We study the cohomology H*(A) = Ext_A(k,k) of a locally finite, connected, cocommutative Hopf algebra A over k = F_p. Specifically, we are interested in those algebras A for which H*(A) is generated as an algebra by H^1(A) and H^2(A). We…

We show that the braided tensor product algebra $A_1\underline{\otimes}A_2$ of two module algebras $A_1, A_2$ of a quasitriangular Hopf algebra $H$ is equal to the ordinary tensor product algebra of $A_1$ with a subalgebra of…

Let R be an integral domain, h non-zero in R such that R/hR is a field, and HA the category of torsionless (or flat) Hopf algebras over R. We call any H in HA "quantized function algebra" (=QFA), resp. "quantized (restricted) universal…

A comodule algebra P over a Hopf algebra H with bijective antipode is called principal if the coaction of H is Galois and P is H-equivariantly projective (faithfully flat) over the coaction-invariant subalgebra B. We prove that principality…

Given a Hopf algebra H, we study modules and bimodules over an algebra A that carry an H-action, as well as their morphisms and connections. Bimodules naturally arise when considering noncommutative analogues of tensor bundles. For…

We introduce a cotwist construction for Hopf algebroids that also entails cotwisting or `quantisation' of the base and which is dual to a previous twisting construction of P. Xu. Whereas the latter applied the construction to the algebra of…

Following the ideas of our previous works math.QA/0008232 (joint with Andruskiewitsch) and math.QA/0101049, we study families of triangular Hopf algebras obtained by twisting finite supergroups by a twist lying entirely in the odd part.…

We develop an Hamiltonian representation of the sl(2,C) algebra on a phase space consisting of N copies of twistors, or bi-spinors. We identify a complete set of global invariants, and show that they generate a closed algebra including…

We study algebraic and homological properties of two classes of infinite dimensional Hopf algebras over an algebraically closed field k of characteristic zero. The first class consists of those Hopf k-algebras that are connected graded as…

We construct three new combinatorial Hopf algebras based on the Loday-Ronco operations on planar binary trees. The first and second algebras are defined on planar trees and labeled planar trees extending the Loday-Ronco and…

We briefly report on our result that, if there exists a realization of a Hopf algebra $H$ in a $H$-module algebra $A$, then their cross-product is equal to the product of $A$ itself with a subalgebra isomorphic to $H$ and commuting with…

In this work, we develop systematically the ``Dirichlet Hopf algebra of arithmetics'' by dualizing addition and multiplication maps. We study the additive and multiplicative antipodal convolutions which fail to give rise to Hopf algebra…

We study the structure of the category of graded, connected, countable-dimensional, commutative and cocommutative Hopf algebras over a perfect field $k$ of characteristic $p$. Every $p$-torsion object in this category is uniquely a direct…

We study the Hopf monoid of convex geometries, which contains partial orders as a Hopf submonoid, and investigate the combinatorial invariants arising from canonical characters. Each invariant consists of a pair: a polynomial and a more…

Consider a finite dimensional H-module Lie algebra L over a field of characteristic 0 where H is a Hopf algebra. We prove the analog of Amitsur's conjecture on asymptotic behavior for codimensions of polynomial H-identities of L under some…

A fundamental feature of quantum groups is that many come in pairs of mutually dual objects, like finite-dimensional Hopf algebras and their duals, or quantisations of function algebras and of universal enveloping algebras of Poisson-Lie…

We construct Hopf algebra isomorphisms of discrete multiplier Hopf C*-algebras, and Hopf AF C*-algebras (generalized quantum UHF algebras), from K-theoretical data. Some of the intermediate results are of independent interest, such as a…

In an intriguing paper arXiv:math/0509083 Khovanov proposed a generalization of homological algebra, called Hopfological algebra. Since then, several attempts have been made to import tools and techiniques from homological algebra to…