My research activity has been concentrated mainly in the representation theory of groups and algebras. My first achievements were in the field of integral representations of orders. Namely, I gave a criterion for a cubic order being of finite representation type; then, in cooperation with A.Roiter, I spread this criterion to all commutative orders, together with the complete list of indecomposable representations for the orders of finite type. Further, in cooperation with V.Kirichenko, I gave a classification of the so called Bassian orders, proved a reduction theorem for a wider class of quasi-Bassian orders and, using these results, obtained a criterion for any primary order being of finite representations type. At the same time, I developed the applications of the idèles technique for study of genera of integral representations, in particular, proved the analogues of the theorems on the distribution of prime ideals for the distribution of maximal submodules in a genus.
In the next period I mainly studied the classification problems of linear algebra in connection with the representation theory of finite dimensional algebras. I gave a general framework for studying such classification problems in terms of the categories of matrices, widely used till now. I also generalized the theorem of Gabriel-Bernstein-Gelfand-Ponomarev on representations of quivers for representations of partially ordered sets and gave an interpretation of these representations in terms of representations of a special class of orders. At the same time I considered a distinction between two classes of classification problems, further called ``tame'' and ``wild'' ones, and gave the first example showing, for representations of commutative algebras, any problem is either tame or wild. Later on, I proved this result for the general case of ``representations of boxes,'' in particular, for categories of matrices and for representations of any finite dimensional algebras. In cooperation with V.Bondarenko, I also gave an explicit criterion for a modular group algebra to be tame (or wild). Together with S.Ovsienko, I considered coverings of algebras and matrix problems and proved that the representation type (finite, tame, wild) is preserved under coverings. Moreover, we gave an explicit correspondence between representations of an algebra and its covering in tame case. For an account on this topic see my survey (Download PDF). Later on I extended this technique so that it could be aplied to derived categories and proved the tame-wild dichotomy for derived categories of finite dimensional algebras (in collaboration with V.Bekkert). (Download PDF)
I was also working in the general theory of rings and modules, in particular, in the theory of commutative rings and of classical orders. Here I gave structure theorems for hereditary rings and modules over them, for serial rings, for locally conjugate orders, for the semi-group of ideals of commutative Noetherian rings. I also gave necessary and sufficient conditions for the existence of maximal orders in an algebra and started a common study of lattices over the orders of arbitrary Krull dimension (in cooperation with L.Chernova).
At about this time, I organized in Kiev University a working group on representation theory, which consisted mainly of young mathematicians (partially, my graduate students). The aim of it was to unify the people working in ``non-classical'' areas of representation theory, such as integral representations, representations of non-semi-simple algebras, etc., and to spread our activity to new areas, where our methods could be used.
Further, I started a study of representations of Lie algebras and Lie groups. It was a new topic for Kiev University, so I tried to organize a group of younger researchers working in this area. First, I studies weight representations of Lie algebra SL(2), in particular, gave a classification of the restricted representations over a field of positive characteristic. Later, I was working in this area mainly in cooperation with V.Futorny and S.Ovsienko. We introduced a new class of representations of Lie algebras SL(n), the so called ``Gelfand-Zetlin modules,'' and proved structure theorems for them. We also generalized the known Harish-Chandra homomorphism and used it to obtain information about generalized Verma modules. At the same time I used the technique of classification problems of linear algebra to study representations of ``mixed'' Lie groups (i.e., neither reductive, nor solvable) and described a wide class of representations (those of ``general position'') for linear groups over Dynkin algebras. In collaboration with V.Mazorchuk I established representation types of some categories of relative Harish-Chandra modules. We also studies a new sort Koszul duality for modules over quasihereditary algebras, especially of those arising from representations of Lie algebras.
Starting from 1991, I also returned to my activity related to integral representations, or, more precise, to the theory of Cohen-Macaulay modules, mainly, in Krull dimension 1. That was done in collaboration with G.-M.Greuel. First, we spread to this situation the ``tame-wild'' dichotomy. Then we studied Cohen-Macaulay modules over local rings of singular curves, gave an explicit criterion for their classification to be tame (or wild) and recovered the relations with the classification of plane curve singularities, namely, with unimodal singularities. We also constructed versal families (with projective bases) of Cohen-Macaulay modules of prescribed rank and used them to prove a semi-continuity theorem for the number of parameters defining such modules. To consider Cohen-Macaulay algebras of Krull dimension 2, we had to classify vector bundles over projective curves. We proved that such a classification is always either of finite type, or tame, or wild (in the same sense as in the theory of finite dimensional algebras) and gave a complete description of curves of finite and tame types, as well as of all vector bundles over such curves. Using this criterion, we proved that among minimal elliptic surface singularities, only simple elliptic and cusp singularities are tame with respect to the classification of Cohen-Macaulay modules. (See my surveys of 1994 (Download PDF) and 2002 (Download PDF).) Further, in collaboration with my student I.Burban on I applied this technique to study derived categories of modules and coherent sheaves. Especially, we got a complete description of derived categories of coherent sheaves over singular curves of finite and tame vector bundle types. We also introduced a new class of rings that are non-commutative analogues of locale rings of simple double points and described derived categories of modules over such rings. (Download PDF of a survey)
At last, in collaboration with H.Baues, I applied the developed methods to the classification problems of homotopy types of polyhedra. We gave a classification of stable homotopy types of the (n-1)-connected polyhedra with torsion free homology of dimension n+d for d=4,5 and proved that for d>5 there is infinitely many stable homotopy classes of such polyhedra. (Download PDF) Then I classified their stable homotopy types for d=6 and proved that for d>6 this classification problem is wild. (Download PDF) We also studied homotopy types of polyhedra having at most two non-trivial homotopy groups. In this connection I also studied polynomial functors playing important role in homotopy theory. I gave a complete description of quadratic functors and of several classes of cubic functors, and proved that a complete classification of cubic functors is a wild problem. (Download PDF of a survey)