Izvestiya of Saratov University.
ISSN 1816-9791 (Print)
ISSN 2541-9005 (Online)


Lie algebra

Non-reductive spaces with equiaffine connections of nonzero curvature

The introduction of this article states the object of our investigation which is structures on homogeneous spaces. The problem of establishing links between the curvature and the structure of a manifold is one of the important problems of geometry. In general, the research of manifolds of various types is rather complicated. Therefore, it is natural to consider this problem in a narrower class of non-reductive homogeneous spaces. If a homogeneous space is reductive, then the space admits an invariant connection.

Cohomology of the Lie Algebra of Vector Fields on Some One-dimensional Orbifold

I. M. Gelfand and D. B. Fuchs have proved that the cohomology algebra of the Lie algebra of vector fields on the unit circle is isomorphic to the tensor product of the polynomial ring with one generator of degree two and the exterior algebra with one generator of degree three. In the present paper the cohomology of the Lie algebra of vector fields on the one-dimensional orbifold S1/Z2 are studied. S1/Z2 is the orbit space under the Z2 group action on the unit circle by reflection in the Ox axis.

Cohomology of Lie algebra of vector fields on S1/Z2

 In the present paper we calculate the diagonal cohomology of Lie algebra of vector fields on S1/Z2 with coefficients in the space of smooth functions and 1-forms, one-dimensional and two-dimensional cohomology with coefficients in R. 

On the A. V. Mikhalev’s Problem for Lie Algebras

Weakened A. V. Mikhalev’ sproblem about the prime radical of artinian Lie algebras is solved.

Three-dimensional Homogeneous Spaces, Not Admitting Invariant Connections

The purpose of the work is the classification of three-dimensional isotropy-faithful homogeneous spaces, not admitting invariant connections. The local classification of homogeneous spaces is equivalent to the description of effective pairs of Lie algebras. If there exists at least one invariant connection then the space is isotropy-faithful, but the isotropy-faithfulness is not sufficient for the space in order to have invariant connections.

Connections of Nonzero Curvature on Three-dimensional Non-reductive Spaces

When a homogeneous space admits an invariant affine connection? If there exists at least one invariant connection then the space is isotropy-faithful, but the isotropy-faithfulness is not sufficient for the space in order to have invariantconnections. If a homogeneousspace is reductive, then the space admits an invariant connection.Thepurposeoftheworkisadescriptionofthree-dimensionalnon-reductivehomogeneousspaces, admitting invariant affine connections of nonzero curvature only, and the affine connections, curvature and torsion tensors.