A Partial Classification of Simple Regular Representations of Bimodules Type \((2,\,2)\) Over the Field of Laurent Series

In this paper, we use Galois descent techniques to find suitable representatives of the regular simple representations of the species of type (2, 2) over \(k_n:= k[\varepsilon ^{1/n}]\), where n is a positive integer and \(k:=\mathbb {C}(\!(\varepsilon )\!)\) is the field of Laurent series over the complexes. These regular representations are essential for the definition of canonical algebras. Our work is inspired by the work done for species of type (1, 4) on k in Geiss and Reynoso-Mercado (Bol. Soc. Mat. Mex. 30(3):87, 2024). We presents all the regular simple representations on the n-crown quiver, and from these, we establish a partial classification of regular simple representations of bimodules type (2, 2).