We prove that the problems of classifying triples of symmetric or skew-symmetric matrices up to congruence, local commutative associative algebras with zero cube radical and square radical of dimension 3, and Lie algebras with central commutator subalgebra of dimension 3 are hopeless since each of them reduces to the problem of classifying pairs of n-by-n matrices up to simultaneous similarity. This is the authors' version of a work that was published in Linear Algebra Appl. 407 (2005) 249-262.

In algebraic geometry over a variety of universal algebras Θ, the group Aut(Θ 0 ) of automorphisms of the category Θ 0 of finitely generated free algebras of Θ is of great importance. In this paper, semi-inner automorphisms are defined for the categories of free (semi)modules and free Lie modules; then, under natural conditions on a (semi)ring, it is shown that all automorphisms of those categories are semi-inner. We thus prove that for a variety R M of semimodules over an IBN-semiring R (an IBN-semiring is a semiring analog of a ring with IBN), all automorphisms of Aut( R M 0 ) are semi-inner. Therefore, for a wide range of rings, this solves Problem 12 left open in [22]; in particular, for Artinian (Noetherian, P I-) rings R, or a division semiring R, all automorphisms of Aut( R M 0 ) are semi-inner. April 30, 2005April 30, . 1991 Mathematics Subject Classification. Primary 16Y60, 16D90,16D99, 17B01; Secondary 08A35, 08C05. Date:Key words and phrases. free module, free semimodule over semiring, free modules over Lie algebras, semi-inner automorphism.

We consider homogeneous varieties of linear algebras over an associative-commutative ring K with 1, i.e., the varieties in which free algebras are graded. Let F = F (x 1 , ..., xn) be a free algebra of some variety Θ of linear algebras over K freely generated by a set X = {x 1 , ..., xn}, End F be the semigroup of endomorphisms of F , and Aut End F be the group of automorphisms of the semigroup End F . We investigate structure of the group Aut End F and its relation to the algebraical and categorical equivalence of algebras from Θ.We define a wide class of R 1 MF-domains containing, in particular, Bezout domains, unique factorization domains, and some other domains. We show that every automorphism Φ of semigroup End F , where F is a free finitely generated Lie algebra over an R 1 MF-domain, is semi-inner. This solves the Problem 5.1 left open in [21]. As a corollary, semi-innerity of all automorphism of the category of free Lie algebras over R 1 MF-domains is obtained. Relations between categorical and geometrical equivalence of Lie algebras over R 1 MFdomains are clarified.The group Aut End F for the variety of m-nilpotent associative algebras over R 1 MF-domains is described. As a consequence, a complete description of the group of automorphisms of the full matrix semigroup of n × n matrices over R 1 MF-domains is obtained.We give an example of the variety Θ of linear algebras over a Dedekind domain such that not all automorphisms of Aut End F are quasi-inner.The results obtained generalize the previous studies of various special cases of varieties of linear algebras over infinite fields.

Let [Formula: see text] be the variety of associative algebras over a field K and A = K 〈x1,…, xn〉 be a free associative algebra in the variety [Formula: see text] freely generated by a set X = {x1,…, xn}, End A the semigroup of endomorphisms of A, and Aut End A the group of automorphisms of the semigroup End A. We prove that the group Aut End A is generated by semi-inner and mirror automorphisms of End A. A similar result is obtained for the automorphism group Aut [Formula: see text], where [Formula: see text] is the subcategory of finitely generated free algebras of the variety [Formula: see text]. The later result solves Problem 3.9 formulated in [17].

This is the authors' version of a work that was published in Linear Algebra Appl. 402 (2005) 135-142. We prove that over an algebraically closed field of characteristic not two the problems of classifying pairs of sesquilinear forms in which the second is Hermitian, pairs of bilinear forms in which the second is symmetric (skew-symmetric), and local algebras with zero cube radical and square radical of dimension 2 are hopeless since each of them reduces to the problem of classifying pairs of n-by-n matrices up to simultaneous similarity.