
TL;DR
This paper constructs a geometric embedding of a translation quiver to demonstrate the existence of an involutive anti-automorphism, leading to explicit characterization of cluster automorphisms for certain cluster algebras.
Contribution
It introduces a novel geometric embedding of translation quivers to explicitly describe involutive automorphisms in cluster algebra theory.
Findings
Existence of an involutive anti-automorphism of the translation quiver $ extbf{Z}Q$.
Explicit characterization of the group of cluster automorphisms for seed $(X,Q)$.
Identification of conditions where $Q$ and $Q^{op}$ are mutation equivalent.
Abstract
We construct a special embedding of the translation quiver in the three-dimensional affine space where is a finite connected acyclic quiver and contains a local slice whose quiver is isomorphic to the opposite quiver of Via this embedding, we show that there exists an involutive anti-automorphism of the translation quiver As an immediate consequence, we characterize explicitly the group of cluster automorphisms of the cluster algebras of seed , where and are mutation equivalent.
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Taxonomy
TopicsAlgebraic structures and combinatorial models · Advanced Topics in Algebra · Advanced Algebra and Geometry
