Ruvim Lipyanski

Problems of classifying associative or Lie algebras and triples of symmetric or skew-symmetric matrices are wild

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.

The problems of classifying pairs of forms and local algebras with zero cube radical are wild

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.

Automorphisms of Categories of Free Modules, Free Semimodules, and Free Lie Modules∗

In algebraic geometry over a variety of universal algebras Θ, the group Aut(Θ0) of automorphisms of the category Θ0 of finitely generated free algebras of Θ is of great importance. In this article, 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 Rℳ of semimodules over an IBN-semiring R (an IBN-semiring is a semiring analog of a ring with IBN), all automorphisms of Aut(Rℳ0) are semi-inner. Therefore, for a wide range of rings, this solves Problem 12 left open in Plotkin (2002); in particular, for Artinian (Noetherian, PI-) rings R, or a division semiring R, all automorphisms of Aut(Rℳ0) are semi-inner.

AUTOMORPHISMS OF THE ENDOMORPHISM SEMIGROUP OF A FREE ASSOCIATIVE ALGEBRA

Let A = A(x 1 , ..., xn) be a free associative algebra in the variety of associative algebras A freely generated over K by a set X = {x 1 , ..., xn}, End A be the semigroup of endomorphisms of A, and Aut End A be the group of automorphisms of the semigroup End A. We investigate the structure of the groups Aut End A and Aut A • , where A • is the category of finitely generated free algebras from A. We prove that the group Aut End A is generated by semi-inner and mirror automorphisms of End F and the group Aut A • is generated by semi-inner and mirror automorphisms of the category A •. This result solves an open Problem formulated in [14].Let be the variety of associative algebras over a field K and A = K 〈x1,…, xn〉 be a free associative algebra in the variety 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 , where is the subcategory of finitely generated free algebras of the variety . The later result solves Problem 3.9 formulated in [17].

Automorphisms of the endomorphism semigroup of a polynomial algebra

Abstract We describe the automorphism group of the endomorphism semigroup End ( K [ x 1 , … , x n ] ) of ring K [ x 1 , … , x n ] of polynomials over an arbitrary field K . A similar result is obtained for automorphism group of the category of finitely generated free commutative–associative algebras of the variety CA commutative algebras. This solves two problems posed by B. Plotkin (2003) [18, Problems 12 and 15] . More precisely, we prove that if φ ∈ Aut End ( K [ x 1 , … , x n ] ) then there exists a semi-linear automorphism s : K [ x 1 , … , x n ] → K [ x 1 , … , x n ] such that φ ( g ) = s ∘ g ∘ s − 1 for any g ∈ End ( K [ x 1 , … , x n ] ) . This extends the result obtained by A. Berzins for an infinite field K .

PROBLEMS OF CLASSIFYING ASSOCIATIVE OR LIE ALGEBRAS OVER A FIELD OF CHARACTERISTIC NOT TWO AND FINITE METABELIAN GROUPS ARE WILD

Let F be a field of characteristic different from 2. It is shown that the problems of classifying (i) local commutative associative algebras over F with zero cube radical, (ii) Lie algebras over F with central commutator subalgebra of dimension 3, and (iii) finite p-groups of exponent p with central commutator subgroup of order p 3

PYTHAGOREAN TRIPLES IN THE UNIFICATION THEORY OF ASSOCIATIVE AND COMMUTATIVE RINGS

ABSTRACT All solutions of Pythagorean equation (P-equation) in relatively free rings of varieties of n-nilpotent associative or commutative-associative rings ( ) are described. In particular, it is shown that Pythagorean equation has no minimal and complete set of solutions in free rings of such varieties. This implies that unification type of these varieties is nullary. It is shown that unification type of P-equation in varieties of associative rings and commutative-associative rings without unit is not finitary. Hence, the unification type of these varieties is also not finitary. We show also that the variety of commutative-associative 3 (or 4)-nilpotent rings of characteristic 2 has nullary unification type.

Pythagorean Triples in Unification Theory of Nilpotent Rings

All solutions of Pythagorean equation (P-equation) x12+ x22 = x32in relatively free rings of varieties of n-nilpotent associative or associative-commutative rings (n=3,4) are described. In particular, it is shown that Pythagorean equation has no minimal and complete set of solutions in free rings of such varieties, so unification type of these varieties is nullary. This is also valid for the variety of associative-commutative 3 (or 4)-nilpotent rings of characteristic two.

On Borel complexity of the isomorphism problems for graph related classes of Lie algebras and finite p-groups

We reduce the isomorphism problem for undirected graphs without loops to the isomorphism problems for a class of finite dimensional

The unification type of the Pythagorean equation in varieties of nilpotent rings

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