Consider a connected, elementary, finite dimensional tame hereditary algebra, thus a path algebra kQ, where k is a field and Q is an acyclic quiver of tame type (i.e. of Euclidean type <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$\widetilde{\mathbb {A}}_{m},\widetilde{\mathbb {D}}_{m},\widetilde{\mathbb {E}}_6,\widetilde{\mathbb {E}}_7,\widetilde{\mathbb {E}}_8$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mover accent="true"> <mi mathvariant="double-struck">A</mi> <mo stretchy="true">~</mo> </mover> <mi>m</mi> </msub> <mo>,</mo> <msub> <mover accent="true"> <mi mathvariant="double-struck">D</mi> <mo stretchy="true">~</mo> </mover> <mi>m</mi> </msub> <mo>,</mo> <msub> <mover accent="true"> <mi mathvariant="double-struck">E</mi> <mo stretchy="true">~</mo> </mover> <mn>6</mn> </msub> <mo>,</mo> <msub> <mover accent="true"> <mi mathvariant="double-struck">E</mi> <mo stretchy="true">~</mo> </mover> <mn>7</mn> </msub> <mo>,</mo> <msub> <mover accent="true"> <mi mathvariant="double-struck">E</mi> <mo stretchy="true">~</mo> </mover> <mn>8</mn> </msub> </mrow> </math></EquationSource> </InlineEquation>). We prove that the factor of a Gabriel-Roiter (GR) inclusion between exceptional modules is also exceptional, unless we are in the Kronecker case (i.e. Q is the Kronecker quiver) with both modules involved in the inclusion being preprojective indecomposable. In this way we extend a result of Bo Chen in [<CitationRef CitationID="CR1">1</CitationRef>] valid in the cases <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$\widetilde{\mathbb {A}}_m,\widetilde{\mathbb {D}}_{m}$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mover accent="true"> <mi mathvariant="double-struck">A</mi> <mo stretchy="true">~</mo> </mover> <mi>m</mi> </msub> <mo>,</mo> <msub> <mover accent="true"> <mi mathvariant="double-struck">D</mi> <mo stretchy="true">~</mo> </mover> <mi>m</mi> </msub> </mrow> </math></EquationSource> </InlineEquation> and for k algebraically closed.

Factors of Gabriel-Roiter inclusions in tame cases

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2025年10月31日

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Szántó Csaba

摘要

Consider a connected, elementary, finite dimensional tame hereditary algebra, thus a path algebra kQ, where k is a field and Q is an acyclic quiver of tame type (i.e. of Euclidean type $\widetilde{\mathbb {A}}_{m},\widetilde{\mathbb {D}}_{m},\widetilde{\mathbb {E}}_6,\widetilde{\mathbb {E}}_7,\widetilde{\mathbb {E}}_8$ ${\tilde{A}}_{m}, {\tilde{D}}_{m}, {\tilde{E}}_{6}, {\tilde{E}}_{7}, {\tilde{E}}_{8}$ ). We prove that the factor of a Gabriel-Roiter (GR) inclusion between exceptional modules is also exceptional, unless we are in the Kronecker case (i.e. Q is the Kronecker quiver) with both modules involved in the inclusion being preprojective indecomposable. In this way we extend a result of Bo Chen in [1] valid in the cases $\widetilde{\mathbb {A}}_m,\widetilde{\mathbb {D}}_{m}$ ${\tilde{A}}_{m}, {\tilde{D}}_{m}$ and for k algebraically closed.