Coclass theory for nilpotent associative algebras

Abstract

Coclass theory for finite p-groups was introduced by Leedham-Green & Newman in 1980 and has become a very fruitful approach in the investigation of finite p-groups yielding many interesting results. The central aim of this thesis is to initiate a coclass theory for nilpotent associative algebras over fields and hence to gain further insight into their structure. An essential tool in our investigation are the coclass graphs associated to the nilpotent associative F-algebras of a fixed coclass r. We consider several important features of these graphs. We give a complete structure description of the inverse limits associated to the infinite paths in the coclass graphs. Using this structure description we prove that the number of equivalence classes of infinite paths in a coclass graph is finite if and only if r is less or equal than one or the underlying field is finite. A coclass graph is the disjoint union of its maximal descendant trees. We prove that the root of a maximal descendant tree in a coclass graph for coclass r has dimension bounded by 2r. Each maximal descendant tree can contain either zero, one or several infinite paths starting at its root. A coclass tree is defined to contain exactly one infinite path starting at its root. It is shown that each maximal coclass tree has bounded depth, i.e. there is a non-negative integer c, such that for every algebra in the maximal coclass tree the distance to an algebra on the unique infinite path is at most c. We prove that for finite fields a coclass graph consists of finitely many maximal coclass trees and finitely many other vertices. This made coclass graphs over finite fields accessible for computational experiments using the algorithms we present. The striking observation in the experimental data is that all coclass trees in our examples exhibit a periodic pattern. Based on this observation we conjecture that for a fixed finite field and a fixed non-negative integer r all algebras in the associated coclass graphs can be described by finitely many parametrized presentations. We prove this periodicity conjecture for r=0 and r=1.