On closed subsets of roots and cohomology of regular subalgebras

This article presents a method of describing the closed subsets of a root system. For example, determining the closed subsets of a root system of type $A_l$ is equivalent to introducing a quasi-ordering on a set of $l+1$ elements. It is shown that the dimensions of the cohomology groups of the corresponding regular subalgebras of $\mathfrak{gl}(l,K)$ in the adjoint representation can be expressed in terms of the Betti numbers of a polyhedron which is constructed by the use of a partial ordering which arises from our quasi-ordering.
Bibliography: 11 titles.