Representation Spaces of the Jordan Plane

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).

A complete description of irreducible components of the representation variety mod(R, n) is obtained for any dimension n, it is shown that the representation variety is equidimensional.

We investigate the influence of the property of the noncommutative Koszul (or Golod–Shafarevich) complex to be a DG-algebra resolution of an algebra, on the structure of representation spaces. It is shown that the Jordan plane provides a new example of representational complete intersection (RCI), which is not a preprojective algebra.

Key Words:

• Golod–Shafarevich complex
• Irreducible components
• (noncommutative complete intersections) NCCI
• Representation spaces
• (representational complete intersections) RCI

2010 Mathematics Subject Classification:

• Primary 16G30, 16G60, 16D25
• Secondary 16A24

ACKNOWLEDGMENTS

The work on this circle of questions was started during my visit at the Max–Planck–Intitut für Mathematik in Bonn. The content of the first 9 chapters appeared as an MPI preprint [19]. I am thankful to this institution for support and hospitality and to many colleagues with whom I have been discussing ideas related to this paper.

I also would like to greatly acknowledge the support of the ERC grant COIMBRA and of the ETF9038 grant of the Estonian Research Council.

Notes

Communicated by A. Smoktunowicz.