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

The class of $N$-Koszul graded algebras of finite global dimension has gained lots of attention in recent years, especially in the study of Artin-Schelter regular algebras. While structurally rich and concrete, the only known examples of…

We present new infinite families of expander graphs of vertex degree 4, which is the minimal possible degree for Cayley graph expanders. Our first family defines a tower of coverings (with covering indices equals 2) and our second family is…

We show that if $k$ is a countable field, then there exists a finitely generated, infinite-dimensional, primitive algebraic $k$-algebra $A$ whose Gelfand-Kirillov dimension is at most six. In addition to this we construct a two-generated…

Smoktunowicz, Lenagan, and the second-named author recently gave an example of a nil algebra of Gelfand-Kirillov dimension at most three. Their construction requires a countable base field, however. We show that for any field $k$ and any…

Let $X$ be a finitely generated left module over a left artinian ring $R$, and let $p(X)=\{l_i\}$ be the infinite sequence of nonnegative integers where $l_i$ is the length of the $i$-th term of the minimal projective resolution of $X$. We…

We classify simple representations of Khovanov-Lauda-Rouquier algebras in finite type. The classification is in terms of a standard family of representations that is shown to yield the dual PBW basis in the Grothendieck group. Finally, we…

We give two kinds of generalizations of Arakelov type inequalities for higher dimensional families. These results give higher dimensional generalizations (in both fibers and bases) of the weakly boundedness in Par\v{s}in-Arakelov's…

Let $A$ be a finite dimensional algebra over a field $K$ with enveloping algebra $A^e=A^{op} \otimes_K A$. We call algebras $A$ that have the property that the subcategory of Gorenstein projective modules in $mod-A$ coincide with the…

Let $k$ be a field of characteristic $0$. Using the method of idealization, we show that there is a non-Koszul, quadratic, Artinian, Gorenstein, standard graded $k$-algebra of regularity $3$ and codimension $8$, answering a question of…

We call a finite-dimensional K-algebra A geometrically irreducible if for all d all connected components of the affine scheme of d-dimensional A-modules are irreducible. We prove that a geometrically irreducible algebra with exactly two…

We study algebras k[x_1,...,x_n]/I which admit a grading by a subsemigroup of N^d such that every graded component is a one-dimensional k-vector space. V.I.~Arnold and coworkers proved that for d = 1 and n <= 3 there are only finitely many…

Shafarevich conjecture/problem is about the finiteness of isomorphism classes of a family of varieties defined over a number field with good reduction outside a finite collection of places. For K3 surfaces, such a finiteness result was…

We consider algebras defined from quivers with relations that are k-th order derivations of a superpotential, generalizing results of Dubois-Violette to the quiver case. We give a construction compatible with Morita equivalence, and show…

This paper generalizes former works of Derksen, Weyman and Zelevinsky about quivers with potentials. We consider semisimple finite-dimensional algebras $E$ over a field $F$, such that $E \otimes_{F} E^{op}$ is semisimple. We assume that $E$…

Differential difference algebras are generalizations of polynomial algebras, quantum planes, and Ore extensions of automorphism type and of derivation type. In this paper, we investigate the Gelfand-Kirillov dimension of a finitely…

The Shafarevich conjecture/problem is about the finiteness of isomorphism classes of a family of varieties defined over a number field with good reduction outside a finite collection of places. For K3 surfaces, such a finiteness result was…

We study maximal subalgebras of an arbitrary finite dimensional algebra over a field, and obtain full classification/description results of such algebras. This is done by first obtaining a complete classification in the semisimple case, and…

We give a description of the connected graded algebras which are finitely generated and presented of global dimension 2 or 3 and which are Gorenstein. These algebras are constructed from multilinear forms. We generalize the construction by…

The dimension algebra of graded groups is introduced. With the help of known geometric results of extension theory that algebra induces all known results of the cohomological dimension theory. Elements of the algebra are equivalence classes…

Let $\Gamma$ be a $T$-ideal of identities of an affine PI-algebra over an algebraically closed field $F$ of characteristic zero. Consider the family $\mathcal{M}_{\Gamma}$ of finite dimensional algebras $\Sigma$ with $Id(\Sigma) = \Gamma$.…