An open subset of the variety of $n$- dimensional algebras, classification and automorphisms

TL;DR

This paper classifies a specific open subset of n-dimensional algebras up to isomorphism, showing they have trivial automorphisms and form an open dense subset in the algebraic variety, with properties under algebraic operations explored.

Contribution

It identifies and characterizes an open dense subset of n-dimensional algebras with trivial automorphisms and analyzes their properties under direct sum and tensor product.

Findings

Algebras in the subset have only trivial automorphisms.

The subset is open and dense in the variety of n-dimensional algebras over algebraically closed fields.

Properties of these subsets under algebraic operations are established.

Abstract

Classification, up to isomorphism, of algebras from a non-empty subset of the variety of $n$- dimensional algebras is presented. It is shown that these algebras have only trivial automorphism and if the basic field is algebraically closed then it is an open dense subset of the variety of $n$- dimensional algebras. Properties of these sets, depending on $n$, with respect to the direct sum and tensor product are considered. Moreover one more way of construction of such subsets with similar properties is considered as well.