Genrich Belitskii

Complexity of matrix problems

In representation theory, the classification problem is called wild if it contains the problem of classifying pairs of matrices up to simultaneous similarity. We show in an explicit form that the last problem contains all classification matrix problems given by quivers or posets. Then we prove that this problem does not contain (but is contained in) the problem of classifying three-valent tensors. Hence, every wild classification problem given by a quiver or poset has the same complexity; moreover, a solution of one of them implies a solution of each of the remaining problems. The problem of classifying three-valent tensors is more complicated.

Normal forms in matrix spaces

For the groupGL(m, C)xGL(n, C) acting on the space ofmxn matrices over C, we introduce a class of subgroups which we call admissible. We suggest an algorithm to reduce an arbitrary matrix to a normal form with respect to an action of any admissible group. This algorithm covers various classification problems, including the “wild problem” of bringing a pair of matrices to normal form by simultaneous similarity. The classical left, right, two-sided and similarity transformations turns out to be admissible. However, the stabilizers of known normal forms (Smiths, Jordans), generally speaking, are not admissible, and this obstructs inductive steps of our algorithm. This is the reason that we introduce modified normal forms for classical actions.

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.

On the normal solvability of cohomological equations on compact topological spaces

The functional equation ϕ(Fx) - ϕ(x) = γ(x) in continuous functions on a compact topological space X is considered, F: X → X is a continuous mapping. It is proved that the equation is normally solvable in C(X) if and only if F is preperiodic, i.e. F p+l = F l for some p ≥ 1, l ≥ 0. The solvability problem in measurable functions is also investigated.

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

Spaces of cohomologies associated with linear functional equations

Let

On solvability of linear difference equations in smooth and real analytic vector functions of several variables

F:X\rightarrow X

A strong version of Implicit Function Theorem

be a

Canonical form of m-by-2-by-2 matrices over a field of characteristic other than two

C^k(X)