Related papers: Ziegler Spectrum and Krull Gabriel Dimension
We investigate degree bounds for fields of rational invariants of representations of finite groups. We prove many cases of a bound for $\mathbb{Z}/p\mathbb{Z}$ conjectured by Blum-Smith, Garcia, Hidalgo, and Rodriguez. For arbitrary groups,…
We study the relationship between $n$-cluster tilting modules over $n$ representation finite algebras and the Euler forms. We show that the dimension vectors of cluster-indecomposable modules give the roots of the Euler form. Moreover, we…
This is the written version of a talk given in memory of Gunnar K\"{a}ll\'{e}n, at the Departments of Theoretical Physics, Physics, and Astronomy of Lund University on February 13, 2009. It will be published in a collection of the papers of…
We present constructive versions of Krull's dimension theory for commutative rings and distributive lattices. The foundations of these constructive versions are due to Joyal, Espan\~ol and the authors. We show that this gives a constructive…
We consider the quantum algebra $U_q(\mathfrak{sl}_2)$ with $q$ not a root of unity. We describe the finite-dimensional irreducible $U_q(\mathfrak{sl}_2)$-modules from the point of view of the equitable presentation.
In this survey we discuss the results on the finitistic dimension of various stratified algebras. We describe what is already known, present some recent estimates, and list some open problems.
We study finite dimensional algebras that appear as fibers of quantum orders over a given point of variety of center. We present the formula for the number of irreducible representations and check it for it for the algebra of twisted…
The Gelfand--Kirillov dimension has gained importance since its introduction as an tool in the study of non-commutative infinite dimensional algebras and their modules. In this paper we show a dichotomy for the Gelfand--Kirillov dimension…
These are the notes for a course on representations of quivers for second year students in Paderborn in summer 2007. My aim was to provide a basic introduction without using any advanced methods. It turns out that a good knowledge of linear…
In this note we present a corregindium to our paper "Krull dimension, Overrings and Semistar operations of an integral Domain", Journal of Algebra, 321 (2009), 1497--1509.
In this note, we explore some recent advancements in enumerative algebraic geometry, focusing particularly on the role of quantum K-theory of quiver varieties as viewed through the lens of integrable systems. We highlight a number of…
We show that Nichols algebras of most simple Yetter-Drinfeld modules over the projective special linear group over a finite field, corresponding to non-semisimple orbits, have infinite dimension. We spell out a new criterium to show that a…
We measure the spectral dimension of universes emerging from nonperturbative quantum gravity, defined through state sums of causal triangulated geometries. While four-dimensional on large scales, the quantum universe appears two-dimensional…
These notes expand a four-hour lecture course given in Heidelberg in March 2023, as part of the "Spring School on non-Archimedean Geometry and Eigenvarieties". They are designed for graduate students and other learners. We introduce Huber…
Differential difference algebras were introduced by Mansfield and Szanto, which arose naturally from differential difference equations. In this paper, we investigate the Gelfand-Kirillov dimension of differential difference algebras. We…
Extra dimensions provide a new window on a number of problems faced by the Standard Model. The following provides an introduction to this very broad subject aimed at experimental graduate students and post-docs based on a lecture given at…
These are lecture notes from author's mini-course during Session 1: "Vertex algebras, W-algebras, and application" of INdAM Intensive research period "Perspectives in Lie Theory", at the Centro di Ricerca Matematica Ennio De Giorgi, Pisa,…
We establish a lower bound for the representation dimension of all the classical Hecke algebras of types A, B and D. For all the type A algebras, and most of the algebras of types B and D, we also establish upper bounds. Moreover, we…
We relate the notions of spectral gap for unitary representations and subfactors with definability of certain important sets in the corresponding structures. We give several applications of this relationship.
This is a brief survey on the current state of play on the programs to classify infinite dimensional Hopf algebras of Gel'fand-Kirillov dimension one or two. We list a number of open questions and suggest directions for future work.