Related papers: Introducing Crystalline Graded Algebras
We construct a graded cluster algebra structure on the Cox ring of a smooth complex variety $Z$, depending on a base cluster structure on the ring of regular functions of an open subset $Y$ of $Z$. After considering some elementary examples…
We consider an orbit category of the bounded derived category of a path algebra of type A_n which can be viewed as a -(m+1)-cluster category, for m >= 1. In particular, we give a characterisation of those maximal m-rigid objects whose…
Generalized Cluster Algebras (GCA) are generalizations of Cluster Algebras (CA) with higher-order exchange relations. Previously, Chekhov-Shapiro conjectured that every GCA can be embedded into a CA. In this paper, we prove a modified…
We study the structural and homological properties of graded Artinian modules over generalized Weyl algebras (GWAs), and this leads to a decomposition result for the category of graded Artinian modules. Then we define and examine a category…
Generalized Weyl Algebras (GWAs) appear in diverse areas of mathematics including mathematical physics, noncommutative algebra, and representation theory. We study the invariants of quantum GWAs under finite automorphisms. We extend a…
The first Weyl algebra over $k$, $A_1 = k \langle x, y\rangle/(xy-yx - 1)$ admits a natural $\mathbb{Z}$-grading by letting $\operatorname{deg} x = 1$ and $\operatorname{deg} y = -1$. Paul Smith showed that $\operatorname{gr}- A_1$ is…
We consider the groups of regular circulant matrices over finite fields and integer residue class rings. In both cases we present a formula for the order of these groups. We also make a first step towards finding the algebraic structure of…
In connection with generalized cluster algebras we introduce a certain generalization of the celebrated Rogers dilogarithm, which we call the Rogers dilogarithms of higher degree. We show that there is an identity of these generalized…
Leavitt path algebras associate to directed graphs a $\mathbb Z$-graded algebra and in their simplest form recover the Leavitt algebras $L(1,k)$. In this note, we first study this $\mathbb Z$-grading and characterize the ($\mathbb…
In the note some construction of Lie algebras is introduced. It is proved that the construction has the same property as a well known wreath product of groups [1]: Any extension of groups can be embedded into their wreath product [2].
We introduce new objects, called $(G,c)$-bands, associated with a simple simply-connected algebraic group $G$, and a Coxeter element $c$ in its Weyl group. We show that bands of a given type are the $K$-points of an infinite dimensional…
We introduce the class of epsilon-strongly graded rings and show that it properly contains both the class of strongly graded rings and the class of unital partial crossed products. We determine precisely when an epsilon-strongly graded ring…
We develop the notion of crystal in the context of derived algebraic geometry, and to connect crystals to more classical objects such as D-modules.
In various subjects including mathematics, one can hope to use mathematical thinking well when the right kinds of algebraic structure to consider can be discovered or spotted. Therefore, it would help to understand kinds of algebraic…
We generalize a construction of Bell and Rogalski to realize new examples of $\mathbb{Z}^n$-graded simple rings. This construction also generalizes TGWAs of type $(A_1)^n$. In addition to considering basic properties of these algebras, we…
We study graded twisted tensor products and graded twists of twisted generalized Weyl algebras (TGWAs). We show that the class of TGWAs is closed under these operations assuming mild hypotheses. We generalize a result on cocycle equivalence…
We give an introduction to the theory of cluster categories and cluster tilted algebras. We include some background on the theory of cluster algebras, and discuss the interplay with cluster categories and cluster tilted algebras.
This is an overview of higher structural constructions in physics. The main motivations of our current attempt are as follows: (i) to provide a brief introduction to derived algebraic geometry, (ii) to understand how derived objects…
An involutive Lie bialgebra induces a Batalin-Vilkovisky operator on its exterior algebra. We introduce a graded generalization of the necklace Lie bialgebra, which depends on a choice of a quiver $Q$. We relate the resulting…
As in many other fields, the rapid rise of generative artificial intelligence is reshaping materials discovery by offering new ways to propose crystal structures and, in some cases, even predict desired properties. This review provides a…