The first Chevalley–Eilenberg Cohomology group of the Lie algebra on the transverse bundle of a decreasing family of foliations

In [L. Lebtahi, Lie algebra on the transverse bundle of a decreasing family of foliations, J. Geom. Phys. 60 (2010), 122–133], we defined the transverse bundle VkVk to a decreasing family of kk foliations FiFi on a manifold MM. We have shown that there exists a (1,1)(1,1) tensor JJ of VkVk such that Jk≠0Jk≠0, Jk+1=0Jk+1=0 and we defined by LJ(Vk)LJ(Vk) the Lie Algebra of vector fields XX on VkVk such that, for each vector field YY on VkVk, [X,JY]=J[X,Y][X,JY]=J[X,Y].In this note, we study the first Chevalley–Eilenberg Cohomology Group, i.e. the quotient space of derivations of LJ(Vk)LJ(Vk) by the subspace of inner derivations, denoted by H1(LJ(Vk))H1(LJ(Vk)).