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 <InlineEquation ID="IEq3"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="10468_2025_10327_Article_IEq3.gif" Format="GIF" Height="21" Rendition="HTML" Resolution="72" Type="Linedraw" Width="89" /> </InlineMediaObject> <EquationSource Format="TEX">$k_n:= k[\varepsilon ^{1/n}]$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>k</mi> <mi>n</mi> </msub> <mo>:</mo> <mo>=</mo> <mi>k</mi> <mrow> <mo stretchy="false">[</mo> <msup> <mi>ε</mi> <mrow> <mn>1</mn> <mo stretchy="false">/</mo> <mi>n</mi> </mrow> </msup> <mo stretchy="false">]</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation>, where n is a positive integer and <InlineEquation ID="IEq4"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="10468_2025_10327_Article_IEq4.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="75" /> </InlineMediaObject> <EquationSource Format="TEX">$k:=\mathbb {C}(\!(\varepsilon )\!)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>k</mi> <mo>:</mo> <mo>=</mo> <mi mathvariant="double-struck">C</mi> <mo stretchy="false">(</mo> <mspace width="-0.166667em" /> <mo stretchy="false">(</mo> <mi>ε</mi> <mo stretchy="false">)</mo> <mspace width="-0.166667em" /> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation> 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, <CitationRef CitationID="CR7">2024</CitationRef>). 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).

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

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2025年3月26日

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Hernán Giraldo,
David Reynoso-Mercado,
Pedro Rizzo

摘要

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 $k_n:= k[\varepsilon ^{1/n}]$ $k_{n} : = k [ε^{1 / n}]$ , where n is a positive integer and $k:=\mathbb {C}(\!(\varepsilon )\!)$ $k : = C ((ε))$ 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).

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

摘要