arXiv subjects

Ibrahim Saleh

Publications and source records attributed to Ibrahim Saleh.

7 recordsLinked to original sources

arXiv2026

On the Mutations of Gentle Quivers and Admissible Ideals

Fomin Zelevinsky quiver mutation is an essential tool in the theory of cluster algebras. In this paper, we extend this framework to quivers with relations by introducing an involutive mutation operation on admissible ideals of path algebras. Briefly our mian motive, is to develop systematic framework for producing new admissible and gentle bound quiver algebras from given ones. So, for every cluster structure we associate classes of bound and gentle algebras. Finally we also suggest several directions for future work, including the study of mutation classes of gentle algebras, the relation with derived equivalence, and the possibility of developing a broader decorated mutation theory for bound quivers with relations.

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arXiv2024

Rooted mutation groups and finite type cluster algebras

For a fixed seed $(X, Q)$, a \emph{rooted mutation loop} is a sequence of mutations that preserves $(X, Q)$. The group generated by all rooted mutation loops is called \emph{rooted mutation group} and will be denoted by $\mathcal{M}(Q)$. The \emph{global mutation group} of $(X, Q)$, denoted $\mathcal{M}$, is the group of all mutation sequences subject to the relations on the cluster structure of $(X, Q)$. In this article, we show that two finite type cluster algebras $\mathcal{A}(Q)$ and $\mathcal{A}(Q')$ are isomorphic if and only if their rooted mutation groups are isomorphic and the sets $\mathcal{M}/\mathcal{M}(Q)$ and $\mathcal{M'}/\mathcal{M}(Q')$ are in one to one correspondence. The second main result shows that the group $\mathcal{M}(Q)$ and the set $\mathcal{M}/\mathcal{M}(Q)$ determine the finiteness of the cluster algebra $\mathcal{A}(Q)$ and vice versa.

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arXiv2024

Symmetric mutations algebras in the context of sub-cluster algebras

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 $σ(Q)$, for some permutation $σ$. The subalgebra of $\mathcal{A}(Q)$ generated by all symmetric cluster variables is called the \emph{symmetric mutation subalgebra} and is denoted by $\mathcal{B}(Q)$. In this paper we identify the class of cluster algebras that satisfy $\mathcal{B}(Q)=\mathcal{A}(Q)$, which contains almost every quiver of finite mutation type. In the process of proving the main theorem, we provide a classification of quivers mutation classes based on their weights. Some properties of symmetric mutation subalgebras are given.

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arXiv2023

On The Mutations Loops of Valued Quivers

A mutation loop of a valued quiver $Q$, is a combination of quiver automorphisms (permutations of vertices and valuations) and mutations that sends $Q$ to itself. In this article we study what we called \emph{global mutations loops} which are sequences formed of mutations only that send each quiver $Q'$ in the mutation class $[Q]$ to $σ(Q')$ for some permutation $σ$. Based on the weight of $[Q]$ and the shape of $Q$, we identify which quivers have mutations global loops and we provide them for each case.

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arXiv2017

Clustered Hyperbolic Categories

We introduce a class of categories, called \emph{clustered hyperbolic categories}, which are generated by equivalent categories of representations of some Weyl cluster algebras. Every preseed $p$ gives rise to a \emph{categorical preseed} $\mathcal{P}$ which generates a clustered hyperbolic category through \emph{categorical mutations}. A "categorification" of Weyl cluster algebra is introduced in the sense of defining a map $\mathbf{F}_{p}$ from the clustered hyperbolic category $\mathfrak{C}(\mathcal{P})$ to the Weyl cluster algebra $\mathcal{H}(p)$ where image of $\mathbf{F}_{p}$ generates $\mathcal{H}(p)$.

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arXiv2016

Cluster Structure on Generalized Weyl Algebras

We introduce a class of non-commutative algebras that carry a non-commutative (geometric) cluster structure which are generated by identical copies of generalized Weyl algebras. Equivalent conditions for the finiteness of the set of the cluster variables of these cluster structures are provided. Some combinatorial data, called \textit{cluster strands,} arising from the cluster structure are used to construct irreducible representations of generalized Weyl algebras.

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arXiv2014

Induced Representations of Hopf Algebras

Hopf representation is a module and comodule with a consistency condition that is more general than the consistency condition of Hopf modules. For a Hopf algebra $H$, we construct an induced Hopf representation from a representation of a bialgebra $B$ using a bialgebra epimorphism $π: H\rightarrow B$. Application on the quantum group $E_{q}(2)$ is given.

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