A model for the canonical algebras of bimodules type (1, 4) over truncated polynomial rings

Let \(\mathbb {C}(\!(\varepsilon )\!)\) $\mathbb{C}((\epsilon ))$ be the field of complex Laurent series. We use Galois descent techniques to show that the simple regular representations of the species of type \((1,\, 4)\) $(1,4)$ over \(\mathbb {C}(\!(\varepsilon )\!)\) $\mathbb{C}((\epsilon ))$ are naturally parametrized by the closed points of \(\textrm{Spec}(\mathbb {C}(\!(\varepsilon )\!)[x]){\dot{\cup }}\{1,\,2\}\) $\text{Spec}(\mathbb{C}((\epsilon ))[x])\dot{\cup }\{1,2\}$ . Moreover, we provide weak normal forms for those representations. We use our representatives of the simple regular representations to describe the canonical algebras associated to the species of type (1, 4) over \(\mathbb {C}(\!(\varepsilon )\!)\) $\mathbb{C}((\epsilon ))$ . This suggests a model of those algebras in the sense of the work of Geiss et al. (Invent Math 209(1):61–158, 2017; Math Z 295:1245–1277, 2020).