Related papers: Golod-Shafarevich type theorems and potential alge…

An infinite filiform Lie algebra L is residually nilpotent and its graded associated with respect to the lower central series has smallest possible dimension in each degree but is still infinite. This means that gr(L) is of dimension two in…

We consider a Sklyanin algebra S with 3 generators, which is the quadratic algebra over a field k with three generators x,y,z given by three relations pxy+qyx+rzz=0, pyz+qzy+rxx=0 and pzx+qxz+ryy=0, where p,q,r are parameters from the…

The main goal of this paper is to study the class of algebras for which the global dimension of the endomorphism ring of the generator-cogenerator, given by the sum of the projective and injective modules, is equal to three. We will refer…

We give an algebraic construction of standard modules (infinite dimensional modules categorifying the PBW basis of the underlying quantized enveloping algebra) for Khovanov-Lauda-Rouquier algebras in all finite types. This allows us to…

We prove the Shafarevich conjecture for very irregular varieties of dimension less than half the dimension of their Albanese variety, subject to some mild numerical conditions. Our proof relies on the Lawrence-Venkatesh method as in the…

Let $\mathfrak g = \mathfrak{gl}_N(k)$, where $k$ is an algebraically closed field of characteristic $p > 0$, and $N \in \mathbb Z_{\ge 1}$. Let $\chi \in \mathfrak g^*$ and denote by $U_\chi(\mathfrak g)$ the corresponding reduced…

Let $H_8$ be the neither commutative nor cocommutative semisimple eight dimensional Hopf algebra, which is also called Kac-Paljutkin algebra \cite{MR0208401}. All simple Yetter-Drinfel'd modules over $H_8$ are given. As for simple objects…

Let $\kk$ be a field, $R$ a standard graded quadratic $\kk$-algebra with $\dim_{\kk}R_2\le 3$, and let $\ov\kk$ denote an algebraic closure of $\kk$. We construct a graded surjective Golod homomorphism $\varphi \colon P\to…

We establish the geometric Shafarevich boundedness conjecture for the moduli stack of stable minimal models, including in particular the moduli stack of KSB pairs.

We obtain a polynomial upper bound in the finite-field version of the multidimensional polynomial Szemer\'{e}di theorem for distinct-degree polynomials. That is, if $P_1, ..., P_t$ are nonconstant integer polynomials of distinct degrees and…

Let G be a finite group and A a finite dimensional G-graded algebra over a field of characteristic zero. When A is simple as a G-graded algebra, by mean of Regev central polynomials we construct multialternating graded polynomials of…

This survey presents recent developments concerning the Shafarevich conjecture, non-abelian Hodge theories, hyperbolicity, and the topology of complex algebraic varieties, as well as the interplay among these areas. More precisely, we…

In this paper we derive estimates for linear potentials that hold away from thin subsets. And, inspired by the celebrated work of Huber, we verify that, for a subset that is thin at a point, there is always a geodesic that reaches to the…

The Grigorchuk and Gupta-Sidki groups play fundamental role in modern group theory. They are natural examples of self-similar finitely generated periodic groups. The author constructed their analogue in case of restricted Lie algebras of…

For any power series $a(t)$ with exponentially bounded nonnegative integer coefficients we suggest a simple construction of a finitely generated monomial associative algebra $R$ with Hilbert series $H(R,t)$ very close to $a(t)$. If $a(t)$…

Koszul algebras have arisen in many contexts; algebraic geometry, combinatorics, Lie algebras, non-commutative geometry and topology. The aim of this paper and several sequel papers is to show that for any finite dimensional algebra there…

We introduce a very natural topology on the set of total orderings of monomials of any algebra having a countable basis over a field. This topological space and some notable subspaces are compact. This topological framework allows us to…

Let $G$ be a semisimple algebraic group over an algebraically closed field $k$ of positive characteristic $p$. Under some restrictions on the size of $p$, the present paper establishes new results on the $G$-module structure of…

We classify $n$-representation infinite algebras $\Lambda$ of type \~A. This type is defined by requiring that $\Lambda$ has higher preprojective algebra $\Pi_{n+1}(\Lambda) \simeq k[x_1, \ldots, x_{n+1}] \ast G$, where $G \leq…

For a finite dimensional algebra $\Lambda$ and a non-negative integer $n$, we characterize when the set $\tilt_n\Lambda$ of additive equivalence classes of tilting modules with projective dimension at most $n$ has a minimal (or…