Related papers: Universal geometric cluster algebras from surfaces
We introduce signed exceptional sequences as factorizations of morphisms in the cluster morphism category. The objects of this category are wide subcategories of the module category of a hereditary algebra. A morphism $[T]:\mathcal A\to…
For a rooted cluster algebra $\mathcal{A}(Q)$ over a valued quiver $Q$, a \emph{symmetric cluster variable} is any cluster variable belonging to a cluster associated with a quiver $\sigma (Q)$, for some permutation $\sigma$. The subalgebra…
We prove that any skew-symmetrizable cluster algebra is unistructural, which is a conjecture by Assem, Schiffler, and Shramchenko. As a corollary, we obtain that a cluster automorphism of a cluster algebra $\mathcal A(\mathcal S)$ is just…
It is often noted that many of the basic concepts of differential geometry, such as the definition of connection, are purely algebraic in nature. Here, we review and extend existing work on fully algebraic formulations of differential…
The main goal of this paper is to investigate the minimal size of families of curves on surfaces with the following property: a family of simple closed curves $\Gamma$ on a surface realizes all types of pants decompositions if for any pants…
We define the notion of universal lift of a projective complex based on non-commutative parameter algebras, and prove its existence and uniqueness. We investigate the properties of parameter algebras for universal lifts.
Algebraic geometry, although little explored in signal processing, provides tools that are very convenient for investigating generic properties in a wide range of applications. Generic properties are properties that hold "almost…
We show that algebraic analogues of universal group covers, surjective group homomorphisms from a $\mathbb{Q}$-vector space to $F^{\times}$ with "standard kernel", are determined up to isomorphism of the algebraic structure by the…
We show that in case a cluster algebra coincides with its upper cluster algebra and the cluster algebra admits a grading with finite dimensional homogeneous components, the corresponding Berenstein-Zelevinsky quantum cluster algebra can be…
We introduce the notion of a lower bound cluster algebra generated by projective cluster variables as a polynomial ring over the initial cluster variables and the so-called projective cluster variables. We show that under an acyclicity…
We study cluster algebras with principal coefficient systems that are associated to unpunctured surfaces. We give a direct formula for the Laurent polynomial expansion of cluster variables in these cluster algebras in terms of perfect…
We study the cluster category of a canonical algebra A in terms of the hereditary category of coherent sheaves over the corresponding weighted projective line X. As an application we determine the automorphism group of the cluster category…
In this paper we explore algebraic and geometric structures that arise on parallelizable manifolds. Given a parallelizable manifold $\mathbb{L}$, there exists a global trivialization of the tangent bundle, which defines a map…
We study cluster algebras with principal and arbitrary coefficient systems that are associated to unpunctured surfaces. We give a direct formula for the Laurent polynomial expansion of cluster variables in these cluster algebras in terms of…
In this article, we introduce the notion of mutation semigroup algebras. This concept simultaneously generalizes cluster algebras and semigroup algebras. We show that, under some mild conditions on the singularities, the spectrum $U={\rm…
We initiate the investigation of representation theory of non-orientable surfaces. As a first step towards finding an additive categorification of Dupont and Palesi's quasi-cluster algebras associated marked non-orientable surfaces, we…
The first part of this thesis deals with certain properties of the quantum symmetric and exterior algebras of Type 1 representations of $U_q(g)$ defined by Berenstein and Zwicknagl. We define a notion of a commutative algebra object in a…
We show that mapping class groups associated to all types of real algebraic curves are virtual duality groups. We also deduce some results about the orbifold homotopy groups of the moduli spaces of real algebraic curves. We achieve these…
This paper investigates the geometry of canonically polarized surfaces defined over a field of positive characteristic which have a nontrivial global vector field, and the implications that the existence of such surfaces has in the moduli…
In the present paper, we introduce two-dimensional categorified Hall algebras of smooth curves and smooth surfaces. A categorified Hall algebra is an associative monoidal structure on the stable $\infty$-category…