Automorphic Equivalence in the Classical Varieties of Linear Algebras

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.