ncts

Let K be a field and let σ be an automorphism of K. Let K⟨X,Y⟩_σ denote the non-commutative K-algebra freely generated by two indeterminates X and Y, the former being σ-linear and the latter being σ⁻¹-linear. Let K[X,Y]_σ denote the quotient of K⟨X,Y⟩_σ by the ideal generated by XY and YX. In this talk, we will explain how the finitely generated left K[X,Y]_σ-modules can be classified by σ-linear representations of a certain family of directed graphs, called "Kraft quivers". When K is a perfect field of characteristic p > 0 and σ is the Frobenius, the algebra K[X,Y]_σ is the mod p Dieudonné ring, and finitely generated left K[X,Y]_σ-modules classify commutative group schemes over K which are killed by p. This case has been studied by Kraft in a German paper from 1975 which has been lost to time. Aside from introducing the graph-theoretic point of view, Kraft's methods were heavily inspired by that of Gelfand and Ponomarev, who studied the case K=C and σ=id in 1968. This talk is the opportunity to briefly review the contents of these papers, over which we claim no originality.