Nilpotent algebras and affinely homogeneous surfaces

TL;DR

This paper explores the structure of finite-dimensional commutative nilpotent algebras, especially those with 1-dimensional annihilators, by associating algebraic hypersurfaces and polynomials, revealing conditions for affine homogeneity.

Contribution

It introduces a novel approach linking nilpotent algebra structures to affine hypersurfaces and nil-polynomials, providing reconstruction methods and characterizing affine homogeneity.

Findings

Algebras with 1-dimensional annihilators relate to affinely equivalent hypersurfaces.

Reconstruction of algebras from hypersurfaces and nil-polynomials is possible.

Not all nilpotent algebras yield affinely homogeneous hypersurfaces.

Abstract

The paper is devoted to the investigation of finite dimensional commutative nilpotent (associative) algebras N over an arbitrary base field of characteristic zero. Due to the lack of a general structure theory for algebras of this type (as opposed to the semi-simple case) we associate various objects to every N which encode the algebra structure. Our main results are in the subclass of algebras having 1-dimensional annihilator, that is, are maximal ideals of Gorenstein algebras of finite vector dimension > 1. Associated structural objects are then, for instance, a class of mutually affinely equivalent algebraic hypersurfaces S in N, and a class of so-called nil-polynomials p, whose degree is the nil-index of N. Then N can be reconstructed from S and even from the quadratic plus cubic part of p. If the algebra N is graded the hypersurface S is affinely homogeneous. The paper closes with an example of an N of dimension 23 and nil-index 5, for which S is not affinely homogeneous.