Dimension vectors of elementary modules of generalized Kronecker quivers

2023/04/09 by Jie Liu, Liu, Jie
Mathematics · #Algebraic structures and combinatorial models #Commutative Algebra and Its Applications #Homotopy and Cohomology in Algebraic Topology

paper · pdf · doi:10.48550/arxiv.2304.04182

Abstract

Let k be an algebraically closed field. The generalized or n-Kronecker quiver K(n) is the quiver with two vertices, called a source and a sink, and n arrows from source to sink. Given a finite-dimensional module M of the path algebra kK(n)=K_n, we consider its dimension vector \underlinedim M=(dim_k M₁, dim_k M₂). Let F=\(x,y)| (2)/(n)x≤ y≤ x\, and let (x,y)\inF. We construct a module X(x,y) of K_n, and we prove it to be elementary. Suppose that \underlinedim M=(x,y). We show that: if M is an elementary module, then x<2n, and when x+y=n+1, the module M is elementary if and only if M is of the form X(x,y).

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