On the stratification and topological structure of classical compact lie groups

The authors realize the stratification of classical connected compact Lie groups. The stratum of the maximal dimension of any such Lie group is a diffeomorphic image of its Lie algebra with respect to the Cayley transform, consisting exactly of all matrices admitting the (inverse) Cayley transform. The further stratification is applied to the subset of exclusive matrices of the Lie group, i. e. the subset of all matrices that do not admit the Cayley transform. The main attention is paid to the Lie groups of unitary matrices. As a consequence, the authors obtained a description of topological structure for the sets of exclusive unitary operators in two-dimensional and three-dimensional complex vector spaces; the first of these sets is realized by physicists as the conformal infinity of the Minkowski space. The stratification of unitary groups uses actions of their Weyl groups on maximal tori and special homogeneous spaces with geometric structures, orbits of canonical unitary matrices with respect to the action of unitary groups by conjugations.