Omel'chenko P. Linearly related sets of self-adjoint operators

This thesis investigates *-algebras, a representation which are pairs of self-adjoint operators in Hilbert space and satisfying a polynomial semilinear and additional relations. Such pairs of operators arise in problems of mathematics and physics, inparticular in the theory of *-representations of quantum groups and Fairlie algebras. The theory of *-representations of involutive algebras, which are discussed in the thesis is closely connected with the theory of *-representations of undirected graphs and the problem of reduction the self-adjoint block matrices graph-defined by the group of unitary block-diagonal matrices, so this problem is considered. Analysis of recent accomplishments and publications shows about the effectiveness of the representation theory of *-algebras in linear spaces over non-commutative fields, in particular the body of quaternions. This theory finds its application in non-relativistic and the relativistic dynamics, field theory and other issues of quantum mechanics. In this regard, some research is dedicated to the study of the structure of linear operators in quaternionic bimodules. Described all irreducible representations of graded analogue of the Lie algebra o(2,H) up to unitary equivalence in quaternionic Hilbert bimodule.