Automorphic Equivalence in the Classical Varieties of Linear Algebras

TL;DR

This paper investigates automorphic equivalence in classical varieties of linear algebras over characteristic zero fields, analyzing automorphism groups of categories of free algebras and their relation to geometric equivalence.

Contribution

It computes the quotient groups of automorphisms over inner automorphisms for various algebra varieties, revealing the structure of automorphic versus geometric equivalence.

Findings

The quotient group is generated by at most two types of strongly stable automorphisms.

Automorphic equivalence does not always imply geometric equivalence, as shown by specific examples.

Automorphisms related to scalar multiplication and element multiplication changes are key to the structure.

Abstract

In this paper we consider some classical varieties of linear algebras over the field which has characteristic 0. For every considered variety we take a category of the finite generated free algebras of this variety. And for every this category we calculate the quotient group of the group of the all automorphisms of this category over the subgroup of the all inner automorphisms. This quotient group measures difference between the geometric equivalence and automorphic equivalence of algebras from this variety. In the all considered varieties of the linear algebras this group is generated by cosets which are presented by no more than two kinds of the strongly stable automorphisms of our category. One kind of automorphisms is connected to the changing of the multiplication by scalar and second one is connected to the changing of the multiplication of the elements of the algebras. We present some examples of the pairs of linear algebras such that the considered automorphism provides the automorphic equivalence of these algebras but these algebras are not geometrically equivalent. These examples are presented for the all considered above varieties of algebras and for both these kinds of the strongly stable automorphisms, when they exist in our group.