On Cubic Functors

Abstract

We prove that the description of cubic functors is a wild problem in the sense of the representation theory. On the contrary, we describe several special classes of such functors (2-divisible, weakly alternative, vector spaces and torsion free ones). We also prove that cubic functors can be defined locally and obtain corollaries about their projective dimensions and torsion free parts.

Key Words:

• Polynomial functors
• Modules
• Matrix problems

Acknowledgments

This research was mainly performed when the author was at the Max-Plank-Institut für Mathematik. I am extremely grateful to the Institute for the excellent opportunity and working conditions. I am also grateful to H.-J. Baues who enthusiastically supported this research and to W. Dreckman for useful discussions. This work was supported by the Max-Planck-Institut für Mathematik and by the CRDF Award UM2-2094.