Deformed cluster maps of type $A_{2N}$

Abstract

We extend recent work of the third author and Kouloukas by constructing deformations of integrable cluster maps corresponding to the Dynkin types $A_{2N}$, lifting these to higher-dimensional maps possessing the Laurent property and demonstrating integrality of the deformations for $N\leq 3$. This provides the first infinite class of examples (in arbitrarily high rank) of such maps and gives information on the associated discrete integrable systems. Key to our approach is a ``local expansion'' operation on quivers which allows us to construct and study mutations in type $A_{2N}$ from those in type $A_{2(N-1)}$.

Figures (8)

• Figure 1: Extension from $Q_{A_{4}}$ to $Q_{A_{6}}$
• Figure 2: Local expansion of the subquiver in $Q_{A_{4}}$
• Figure 3: Type $A_{4}$ deformed quiver
• Figure 4: Quiver corresponding to $\hat{B}_{A_{6}}$
• Figure 5: Mutated quiver $Q'_{A_{6}} = \mu_{3}\mu_{2}\mu_{1}\mu_{15}\mu_{14}\mu_{6} (Q_{A_{6}})$. It has the same structure as Figure \ref{['fig:MuQuiverA6']} with permuted labellings.

Theorems & Definitions (41)

• Definition 2.1
• Example 2.2: Quiver mutation at node 2
• Definition 2.3
• Definition 2.4
• Definition 2.5
• Definition 2.6: Cluster algebra
• Definition 2.7
• Theorem 2.8: Laurent phenomenon
• Definition 2.9: Mutation periodic
• Example 2.10: Type $A_2$