Wednesday, 28 November 2018
Tobias Moede: Nilpotent associative algebras and coclass theory
The coclass of a finite \(p\)-group of order \(p^n\) and class \(c\) is defined as \(n-c\). Using coclass as the primary invariant in the investigation of finite \(p\)-groups turned out to be a very fruitful approach.Together with Bettina Eick, we have developed a coclass theory for nilpotent associative algebras over fields. A central tool are the coclass graphs associated with the algebras of a fixed coclass. The graphs for coclass zero are well understood. We give a full description for coclass one and explore graphs for higher coclasses.
We prove several structural results for coclass graphs, which yield results in the flavor of the coclass theorems for finite \(p\)-groups. The most striking observation in our experimental data is that for finite fields all of these graphs seem to exhibit a periodic pattern. We want to prove and exploit this periodicity in order to describe the infinitely many nilpotent associative \(F\)-algebras of a fixed coclass by finitely many parametrized presentations.
Note that a similar periodicity in the graphs for finite \(p\)-groups has been proved independently by du Sautoy using the theory of zeta functions and by Eick & Leedham-Green using cohomological methods.