Representations of quivers over the algebra of dual numbers
The representations of a quiver Q over a field k (the kQmodules, 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...
Full description
The representations of a quiver Q over a field k (the kQmodules, 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 finitedimensional kalgebras 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 Λmodules, where Λ = k Q [ ϵ ] = k Q [ T ] / 〈 T 2 〉 . The algebra Λ is a 1Gorenstein algebra, thus the torsionless Λmodules are known to be of special interest (as the Gorensteinprojective 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 kQmodules 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 _ .
Saved in:
Format: 
Electronic Article

Language: 
English

Physical Description: 
OnlineRessource

DOI / URN: 
10.1016/j.jalgebra.2016.12.001
