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In this paper, we use Galois descent techniques to find suitablerepresentatives 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 thecomplexes. These regular representations are essential for the definition ofcanonical algebras. Our work is inspired by the work done for species of type$(1,4)$ on $k$ in ``A model for the canonical algebras of bimodules type (1, 4)over truncated polynomial rings''. We presents all the regular simplerepresentations on the $n$-crown quiver, and from these, we establish a partialclassification of regular simple representations of bimodules type $(2,2)$.

A partial classification of simple regular representations of bimodules type $(2,\,2)$ over $\mathbb{C}(\!(\varepsilon)\!)$