Ivan Yudin

A survey on cosymplectic geometry

We give an up-to-date overview of geometric and topological properties of cosymplectic and coKahler manifolds. We also mention some of their applications to time-dependent mechanics.

Examples of compact K-contact manifolds with no Sasakian metric

Using the hard Lefschetz theorem for Sasakian manifolds, we find two examples of compact K-contact nilmanifolds with no compatible Sasakian metric in dimensions 5 and 7, respectively.

Characteristic-free resolutions of Weyl and Specht modules

Abstract We construct explicit resolutions of Weyl modules by divided powers and of co-Specht modules by permutational modules. We also prove a conjecture by Boltje and Hartmann (2010) [7] on resolutions of co-Specht modules.

Topology of 3-cosymplectic manifolds

We continue the programme of Chinea, De Leon and Marrero who studied the topology of cosymplectic manifolds. We study 3-cosymplectic manifolds that are the closest odd-dimensional analogue of hyper-Kahler structures. We show that there is an action of the Lie algebra so(4, 1) on the basic cohomology spaces of a compact 3-cosymplectic manifold with respect to the Reeb foliation. This implies some topological obstructions to the existence of such structures which are expressed by bounds on the Betti numbers. It is known that every 3-cosymplectic manifold is a local Riemannian product of a hyper-Kahler factor and an abelian three-dimensional Lie group. Nevertheless, we present a non-trivial example of a compact 3-cosymplectic manifold that is not the global product of a hyper-Kahler manifold and a flat 3-torus.

A non-Sasakian Lefschetz K-contact manifold of Tievsky type

We find a family of five dimensional completely solvable compact manifolds that constitute the first examples of

Homological properties of quantised Borel-Schur algebras and resolutions of quantised Weyl modules

K

Semiperfect Category-Graded Algebras

-contact manifolds which satisfy the Hard Lefschetz Theorem and have a model of Tievsky type just as Sasakian manifolds but do not admit any Sasakian structure.

Cosymplectic p-spheres

Abstract We continue the development of the homological theory of quantum general linear groups previously considered by the first author. The development is used to transfer information to the representation theory of quantised Schur algebras. The acyclicity of induction from some rank-one modules for quantised Borel–Schur subalgebras is deduced. This is used to prove the exactness of the complexes recently constructed by Boltje and Maisch, giving resolutions of the co-Specht modules for Hecke algebras.

Covariant Lie derivatives and Frolicher-Nijenhuis bracket on Lie algebroids

We introduce the notion of algebras graded over a small category and give a criterion for such algebras to be semiperfect.

Hard Lefschetz Theorem for Vaisman manifolds

Abstract We introduce cosymplectic circles and cosymplectic spheres, which are the analogues in the cosymplectic setting of contact circles and contact spheres. We provide a complete classification of compact 3-manifolds that admit a cosymplectic circle. The properties of tautness and roundness for a cosymplectic p -sphere are studied. To any taut cosymplectic circle on a three-dimensional manifold M we are able to canonically associate a complex structure and a conformal symplectic couple on M × R . We prove that a cosymplectic circle in dimension three is round if and only if it is taut. On the other hand, we provide examples in higher dimensions of cosymplectic circles which are taut but not round and examples of cosymplectic circles which are round but not taut.