Geodesic
On closed subsets of roots and cohomology of regular subalgebras
Sbornik. Mathematics, Tome 33 (1977) no. 1, pp. 125-135 
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/
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     author = {F. M. Malyshev},
     title = {On closed subsets of roots and cohomology of regular subalgebras},
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     language = {en},
     url = {http://geodesic.mathdoc.fr/item/SM_1977_33_1_a6/}
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Voir la notice de l'article provenant de la source Math-Net.Ru

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.

[1] N. Burbaki, Gruppy i algebry Li, izd-vo «Mir», Moskva, 1972 | MR

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[4] I. Morgado, “Note on quasi-orders, partial orders and orders”, Notes comuus mat., 1972, no. 43, 31–40

[5] E. B. Dynkin, “Regulyarnye poluprostye podalgebry poluprostykh algebr Li”, DAN SSSR, 73:5 (1950), 877–880 | MR | Zbl

[6] N. R. Wallack, “On maximal subsystems of root systems”, Canad. J. Math., 20:3 (1968), 555–574 | MR

[7] A. van den Hombergh, “Sur des suites de racines dont la somme des termes est nulle”, Bull. Soc. Math. France, 102:4 (1974), 353–364 | MR | Zbl

[8] G. Hochschild, J.-P. Serre, “Cohomology of Lie algebras”, Ann. Math., 57:3 (1953), 591–603 | DOI | MR | Zbl

[9] G. Leger, E. Luks, “Cohomology theorems for Borel-like solvable Lie algebras in arbitrary characteristic”, Canad. J. Math., 24:6 (1972), 1019–1026 | MR | Zbl

[10] G. Leger, E. Luks, “Cohomology and weight systems for nilpotent Lie algebras”, Bull. Amer. Math. Soc., 80:1 (1974), 77–80 | DOI | MR | Zbl

[11] S. Togo, “On the derivation algebras of Lie algebras”, Canad. J. Math., 13:2 (1961), 201–216 | MR | Zbl