We study the de Rham 1-cohomology H 1 DR (M, G) of a smooth manifold M with values in a Lie group G. By definition, this is the quotient of the set of flat connections in the trivial principal bundle M × G by the so-called gauge equivalence. We consider the case when M is a compact Kähler manifold and G is a solvable complex linear algebraic group of a special class which contains the Borel sub...

By recent work on some conjectures of Pillay, each definably compact group G in a saturated o-minimal expansion of an ordered field has a normal “infinitesimal subgroup” G such that the quotient G/G, equipped with the “logic topology”, is a compact (real) Lie group. Our first result is that the functor G 7→ G/G sends exact sequences of definably compact groups into exacts sequences of Lie group...

If G is a connected compact Lie group, then for almost all prime numbers p the mod p cohomology ring of the classifying space BG is a finitely generated polynomial algebra. In 1961, N. Steenrod [24] asked in general for a determination of all spaces X such that H∗(X,Fp) is a finitely generated polynomial algebra (i.e., such that X has a polynomial cohomology ring); at that time, the only exampl...

There is a well-known theory of differentiable cohomology H diff (G, V ) of a Lie group G with coefficients in a topological vector space V on which G acts differentiably. This is developed by Blanc in [Bl]. It is very desirable to have a theory of differentiable cohomology for a (possibly infinite-dimensional) Lie group G, with coefficients in an arbitrary abelian Lie group A, such that the gr...

We identify the cotangent bundle Lie algebroid of a Poisson homogeneous space G/H of a Poisson Lie group G as a quotient of a transformation Lie algebroid over G. As applications, we describe the modular vector fields of G/H , and we identify the Poisson cohomology of G/H with coefficients in powers of its canonical line bundle with relative Lie algebra cohomology of the Drinfeld Lie algebra as...

We classify the cohomology spaces H(g,K) for all filiform nilpotent Lie algebras of dimension n ≤ 11 over K and for certain classes of algebras of dimension n ≥ 12. The result is applied to the determination of affine cohomology classes [ω] ∈ H(g,K). We prove the general result that the existence of an affine cohomology class implies an affine structure of canonical type on g, hence a canonical...

Hecke operators play an important role in the theory of automorphic forms, and automorphic forms are closely linked to various cohomology groups. This paper is mostly a survey of Hecke operators acting on certain types of cohomology groups. The class of cohomology on which Hecke operators are introduced includes the group cohomology of discrete subgroups of a semisimple Lie group, the de Rham c...

A LieYRep pair consists of a Lie-Yamaguti algebra and its representation. In this paper, we establish the cohomology theory pairs characterize their linear deformations by second group. Then introduce notion relative Rota-Baxter-Nijenhuis structures on pairs, investigate properties, prove that structure gives rise to compatible Rota-Baxter operators under certain condition. Finally, show equiva...