Let ${\mathcal{B}}_{\mathfrak{q}}$ be a finite-dimensional Nichols algebra of diagonal type corresponding to a matrix $\mathfrak{q} \in {\mathbf{k}}^{\theta \times \theta}$. Let ${\mathcal{L}}_{\mathfrak{q}}$ be the Lusztig algebra associated to ${\mathcal{B}}_{\mathfrak{q}}$. We present ${\mathcal{L}}_{\mathfrak{q}}$ as an extension (as braided Hopf algebras) of ${\mathcal{B}}_{\mathfrak{q}}$ by ${\mathfrak Z}_{\mathfrak{q}}$ where ${\mathfrak Z}_{\mathfrak{q}}$ is isomorphic to the universal enveloping algebra of a Lie algebra ${\mathfrak{n}}_\mathfrak{q}$. We compute the Lie algebra ${\mathfrak{n}}_{\mathfrak{q}}$ when $\theta = 2$.

We present a survey of recent developments in the theory of partial actions of groups and Hopf algebras.

In this article we construct three explicit natural subgroups of the Brauer-Picard group of the category of representations of a finite-dimensional Hopf algebra. In examples the Brauer Picard group decomposes into an ordered product of these subgroups, somewhat similar to a Bruhat decomposition. Our construction returns for any Hopf algebra three types of braided autoequivalences and correspondingly three families of invertible bimodule categories. This gives examples of so-called (2-)Morita equivalences and defects in topological field theories. We have a closer look at the case of quantum groups and Nichols algebras and give interesting applications. Finally, we briefly discuss the three families of group-theoretic extensions.

Although multiplier bimonoids in general are not known to correspond to comonoids in any monoidal category, we classify them in terms of maps from the Catalan simplicial set to another suitable simplicial set; thus they can be regarded as (co)monoids in something more general than a monoidal category (namely, the simplicial set itself). We analyze the particular simplicial maps corresponding to that class of multiplier bimonoids which can be regarded as comonoids.

Let $p$ be a prime. In this paper we give a proof of the following result: A valued field $(K,v)$ of characteristic $p > 0$ is $p$-henselian if and only if every element of strictly positive valuation is of the form $x^p - x$ for some $x \in K$.

We give a simple proof that (a generalization of) the complex of injective words has vanishing homology in all except the top degree.

We study the existence of solutions for a new class of integral equations involving a fractional integral with respect to another function. Our techniques are based on the measure of non-compactness concept combined with a generalized version of Darbo's theorem. Some examples are presented to illustrate the obtained results.

A well-known result by M. Min-Oo et al. states that there are no nontrivial basic harmonic $r (0< r<q={\rm codim}\mathcal F)$-forms on a compact foliated Riemannian manifold $(M,\mathcal F)$. We extend this result to a complete foliated Riemannian manifold.