Representations of quivers over the algebra of dual numbers
By Claus Michael Ringel, Pu Zhang. 2011
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Abstract The representations of a quiver Q over a field k (the kQ-modules, where kQ is the path algebra of Q over k) have been studied for a long time, and one knows quite well the structure of the module category mod k Q . It seems to be worthwhile to consider also representations of Q over arbitrary finite-dimensional k-algebras A. Here we draw the attention to the case when A = k [ ϵ ] is the algebra of dual numbers (the factor algebra of the polynomial ring k [ T ] in one variable T modulo the ideal generated by T 2 ), thus to the Λ-modules, where Λ = k Q [ ϵ ] = k Q [ T ] / 〈 T 2 〉 . The algebra Λ is a 1-Gorenstein algebra, thus the torsionless Λ-modules are known to be of special interest (as the Gorenstein-projective or maximal Cohen–Macaulay modules). They form a Frobenius category L , thus the corresponding stable category L _ is a triangulated category. As we will see, the category L is the category of perfect differential kQ-modules and L _ is the corresponding homotopy category. The category L _ is triangle equivalent to the orbit category of the derived category D b ( mod k Q ) modulo the shift and the homology functor H : mod Λ → mod k Q yields a bijection between the indecomposables in L _ and those in mod k Q . Our main interest lies in the inverse, it is given by the minimal L -approximation. Also, we will determine the kernel of the restriction of the functor H to L and describe the Auslander–Reiten quivers of L and L _ .
Subfields: Algebra and Number Theory, Geometry and Topology, Statistical and Nonlinear Physics
View Paper: https://www.semanticscholar.org/paper/974fcdf42d72a731fdbbb1e61ac2f45f2d318dd0
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Citations in the corpus
- Monic representations and Gorenstein-projective modules (2011, background): “For a description of all the pairwise nonisomorphic indecomposable Gorensteinprojective 3-modules see [Ringel and Zhang 2011], where 3 is the path algebra of an arbitrary acyclic quiver over A = k[x]/〈x2〉.”