Representation spaces of the Jordan plane

TL;DR

This paper explores the structure of the representation spaces of the Jordan plane algebra, providing a complete description of their irreducible components and revealing new properties related to non-commutative Koszul complexes.

Contribution

It offers the first detailed analysis of the representation varieties of the Jordan plane and links algebraic properties to geometric structures of these varieties.

Findings

Representation variety $mod(R,n)$ is equidimensional.

Complete description of irreducible components for all dimensions.

Jordan plane exemplifies a new RCI (representational complete intersection).

Abstract

We investigate relations between the properties of an algebra and its varieties of finite-dimensional module structures, on the example of the Jordan plane $R=k<x,y>/ (xy-yx-y^2)$. Complete description of irreducible components of the representation variety $mod (R,n)$ obtained for any dimension $n$, it is shown that the variety is equidimensional. The influence of the property of the non-commutative Koszul (Golod-Shafarevich) complex to be a DG-algebra resolution of an algebra (NCCI), on the structure of representation spaces is studied. It is shown that the Jordan plane provides a new example of RCI (representational complete intersection).