Ovsienko S. Categorical methods for the representation theory

In the thesis have been investigated Harish-Chandra subalgebras and modules, Gelfand-Zetlin modules, quasi-hereditary algebras, boxes and quadratic forms. The notions of Harish-Chandra subalgebra and Harish-Chandra module are investigated. It is proved, that the Gelfand-Zetlin subalgebra in the universal enveloping algebra of the general linear Lie algebra is a Harish-Chandra subalgebra. It is proved, that any character of the Gelfand-Zetlin subalgebra allows finitely many extensions to a simple module of the enveloping algebra. The generalized Harish-Chandra homomorphism is constructed. Conditions of commuting of Ringel duality and Kozhul duality are found. It is proved, that the homological dimension of a quasi-hereditary algebra having a simple preserving duality, equals twice the projective dimension of the characteristic tilting module. The homological algebra in the category of representations of a box is developed. Quasi-hereditary algebras are characterized as Butler-Burt algebras of directedboxes. The existence of an algebra with exact Borel subalgebra in a class of Morita equivalence of a quasi-hereditary algebra is proved. A generalized Ringel duality is constructed. The box technique is extended to derived categories. It is proved, that the coordinates of maximal roots of integral quadratic forms are bounded by 12.