Automorphic equivalence in the classical varieties of linear algebras

Abstract

This research is a continuation of [Tsurkov, Automorphic equivalence of linear algebras, J. Algebra Appl. 13(7) (2014), doi:10.1142/S0219498814500261]. In this paper, we consider some classical varieties of linear algebras over the field [Formula: see text] such that [Formula: see text]. We study the relation between the geometric equivalence and automorphic equivalence of the algebras of these varieties. If we denote by [Formula: see text] one of these varieties, then [Formula: see text] is a category of the finite generated free algebras of the variety [Formula: see text]. In this paper, we calculate for the considered varieties the quotient group [Formula: see text], where [Formula: see text] is a group of all the automorphisms of the category [Formula: see text] and [Formula: see text] is a subgroup of all inner automorphisms of this category. The quotient group [Formula: see text] measures the possible difference between the geometric equivalence and automorphic equivalence of algebras from the variety [Formula: see text]. The results of this paper and [Tsurkov, Automorphic equivalence of linear algebras, J. Algebra Appl. 13(7) (2014), doi: 10.1142/S0219498814500261] are summarized in the table at the end of Sec. 5. We can see from this table that in all considered varieties of the linear algebras the group [Formula: see text] is generated by cosets which are presented by no more than two types of the strongly stable automorphisms of the category [Formula: see text]. The first type of automorphisms is connected with the changing of the multiplication by scalar and a the second type is connected with the changing of the multiplication of the elements of the algebras. In Sec. 6, we present some examples of the pairs of linear algebras such that the considered strongly stable automorphisms provide the automorphic equivalence of these algebras, but these algebras are not geometrically equivalent.