On generic properties of nilpotent algebras

TL;DR

This paper investigates generic properties of finite-dimensional nilpotent algebras, revealing new insights even for classical algebra types, by analyzing algebraic varieties of isomorphism classes and their automorphism groups.

Contribution

It introduces a framework for understanding generic properties of nilpotent algebras using algebraic varieties and explores automorphism groups and ideals in this context.

Findings

Automorphism group of a generic algebra consists of scalar automorphisms modulo $P^2$

Generic ideals are contained in the annihilator of the algebra

Classical nilpotent algebras are graded by degrees with respect to generating sets

Abstract

We study general nilpotent algebras. The results obtained are new even for the classical algebras, such as associative or Lie algebras. We single out certain generic properties of finite-dimensional algebras, mostly over infinite fields. The notion of being generic in the class of $n$-generated algebras of an arbitrary primitive class of $c$-nilpotent algebras appears naturally in the following way. On the set of the isomorphism classes of such algebras one can introduce the structure of an algebraic variety. As a result, the subsets are endowed with the dimensions as algebraic varieties. A subset $Y$ of a set $X$ of lesser dimension can be viewed as negligible in $X$. For example, if $n\gg c$, we determine that an automorphism group of a generic algebra $P$ consists of the automorphisms, which are scalar modulo $P^2$. Generic ideals are in $I(P)$, the annihilator of $P$. In the case of classical nilpotent algebras as above, the generic algebras are graded by the degrees with respect to some generating sets.