We compute the simple finite-dimensional modules and the center of the Drinfeld double of the Jordan plane introduced in [Andruskiewitsch, N. and Peña Pollastri, H. M., “On the restricted Jordan plane in odd characteristic”, J. Algebra Appl. 20, 2140012 (2021). MR4209966] assuming that the characteristic is zero.

This article is devoted to a study of the Hardy space $H^{\log} (\mathbb{R}^d)$ introduced by Bonami, Grellier, and Ky. We present an alternative approach to their result relating the product of a function in the real Hardy space $H^1$ and a function in $BMO$ to distributions that belong to $H^{\log}$ based on dyadic paraproducts. We also point out analogues of classical results of Hardy–Littlewood, Zygmund, and Stein for $ H^{\log}$ and related Musielak–Orlicz spaces.

Relation Gelfand–Tsetlin $\mathfrak{gl}_n$-modules were introduced in [FRZ19], and are determined by some special directed graphs and Gelfand–Tsetlin characters. In this work, we constructed polyhedra associated with the class of relation modules, which includes as a particular case, any classical Gelfand–Tsetlin polytope. Following the ideas presented in [LM04], we give a characterization of $d$-faces of the associated polyhedra in terms of certain tilings and matrices related to the corresponding graph.

Earlier it was described to which extent the Alexander polynomial in several variables of an algebraic link in the Poincaré sphere determines the topology of the link. It was shown that, except some explicitly described cases, the Alexander polynomial of an algebraic link determines the combinatorial type of the minimal resolution of the curve and therefore the topology of the corresponding link. The Alexander polynomial of an algebraic link in the Poincaré sphere coincides with the Poincaré series of the corresponding set of curve valuations. The latter one can be defined as an integral over the space of divisors on the $\mathbb{E}_8$-singularity. Here, we consider a similar integral for rational double point surface singularities over the space of Weil divisors called the Weil–Poincaré series. We show that, except a few explicitly described cases, the Weil–Poincaré series of a collection of curve valuations on a rational double point surface singularity determines the topology of the corresponding link. We give analogous statements for collections of divisorial valuations.

We prove that there exist Cantor sets of arbitrarily large $2$-dimensional Lebesgue measure whose complements admit proper holomorphic embeddings in $\mathbb{C}^2$.

In this paper, we construct open Stein neighbourhoods of compact sets of the form $A \cup K$ in a complex space, where $K$ is a compact holomorphically convex set, $A$ is a compact complex curve with boundary $bA$ of class $\mathscr{C}^2$ which may intersect $K$, and the set $A \cap K$ is $\mathscr{O}(A)$-convex.

We present a direct proof of an Hedenmalm–Shimorin inequality for short antidiagonals proved recently in [HS20, Advances in Mathematics, 2020] and give the three tensor analogue of such inequality.

Let $\mathscr{C}$ be an $n$-exangulated category. In this note, we show that if $\mathscr{C}$ is locally finite, then $\mathscr{C}$ has Auslander–Reiten $n$-exangles. This unifies and extends results of Xiao–Zhu, Zhu–Zhuang, Zhou, and Xie–Lu–Wang for triangulated, extriangulated, $(n+2)$-angulated and $n$-abelian categories, respectively.

We improve both dimension compression and expansion bounds for homeomorphisms with $p$-exponentially integrable distortion. To the first direction, we also introduce estimates for the compression multifractal spectra, which will be used to estimate compression of dimension, and for the rotational multifractal spectra. For establishing the expansion case, we use the multifractal spectra of the inverse mapping and construct examples proving sharpness.

We formulate and prove an analogue of Beurling’s theorem for the Fourier transform on the Heisenberg group. As a consequence, we deduce Hardy and Cowling–Price theorems.