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

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.