Let g be a finite dimensional real Lie bialgebra. We introduce an R -valued cohomology of g for which the space of all inequivalent Lie bialgebra central extensions of g by R is isomorphic to that second cohomology group of g . Furthermore, we study the natural projection of this group on the R -valued Lie algebra second cohomology group of g .

LetG be a finite group and k be a field of characteristic p > 0. A cohomology class ζ ∈ H(G, k) is called productive if it annihilates Ext∗kG(Lζ , Lζ). We consider the chain complexP(ζ) of projective kG-modules which has the homology of an (n−1)-sphere and whose k-invariant is ζ under a certain polarization. We show that ζ is productive if and only if there is a chain map ∆ : P(ζ) → P(ζ) ⊗ P(ζ)...

A cohomology ring algorithm in a dimension-independent framework of combinatorial cubical complexes is developed with the aim of applying it to the topological analysis of high-dimensional data. This approach is convenient in the cup-product computation and motivated, among others, by interpreting pixels or voxels in digital images as cubes. The S-complex theory and so called co-reductions are ...