Geodesic
On closed subsets of roots and cohomology of regular subalgebras
Sbornik. Mathematics, Tome 33 (1977) no. 1, pp. 125-135

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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. 
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     author = {F. M. Malyshev},
     title = {On closed subsets of roots and cohomology of regular subalgebras},
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F. M. Malyshev. On closed subsets of roots and cohomology of regular subalgebras. Sbornik. Mathematics, Tome 33 (1977) no. 1, pp. 125-135. http://geodesic.mathdoc.fr/item/SM_1977_33_1_a6/