On generic properties of nilpotent algebras (2406.15631v1)

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.