2011/01/16 by Gregor Fels, Fels, Gregor, Wilhelm Kaup +1 · 1 citation
Mathematics · #14J70 (Primary) #32V40 (Secondary) #Advanced Algebra and Geometry #Advanced Topics in Algebra #Algebraic structures and combinatorial models #Commutative Algebra (math.AC) #FOS: Mathematics #math.AC #msc:14J70 #msc:32V40
paper · pdf · doi:10.48550/arxiv.1101.3088
27 pages
openalex publication_date 2011/01/16 · arxiv created 2011/08/05 · arxiv updated 2011/08/08 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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.