Cohomology of graded Lie algebras of Cartan type \(S(n,{\mathfrak m})\) (Q909010)

The Lie algebra of Cartan type \(S(n,\bar m)\) is defined as a Lie algebra of special derivations of the divided power algebra of n variables with height \(\bar m=(m_ 1,...,m_ n)\), whose divergence are zero. In this paper 1-cohomology of irreducible modules of the Lie algebra \(S(3,\bar m)\) is calculated. In the restricted case, i.e. \(m_ 1=m_ 2=m_ 3=1\), the restricted 1-cohomology of irreducible representations of \(S(3,\bar m)\) is also calculated. The list of irreducible \(S(3,\bar m)\)-modules M with the property \(H^ 1(S(3,\bar m),M)\neq 0\) is not complete. The case of adjoint module, where \(H^ 1(S(3,\bar m)\), \(S(3,\bar m))\) is \((m_ 1+m_ 2+m_ 3-3)\)-dimensional, is omitted. For other Lie algebras of Cartan type \(L=W(1,\bar m)\), \(W(3,\bar m)\), \(H(2,\bar m)\) and 1-cohomologies of irreducible representations, see the reviewer [Mat. Sb., Nov. Ser. 119(161), 132-149 (1982; Zbl 0504.17006)] and \textit{S. Chiu} and \textit{G. Shen} [Abh. Math. Semin. Univ. Hamb. 57, 139-156 (1987; Zbl 0636.17005)].