Second cohomology of generalized Cartan type \(H\) Lie algebras in characteristic 0 (Q1271000)

The author determines the second cohomology group \(H^2(L,F)\) of the generalized Cartan type \(H\) Lie algebras \(L=:L(A, \delta,\phi)\) under the condition that \(L\) is simple, where \(F\) is a field of characteristic 0 . Let \(M\) be the \(F\)-vector space consisting of all additive maps \(\mu: A \rightarrow F\) such that \(\mu(\delta_1)=\cdots =\mu(\delta_n)\), and \(\mu(\delta_1)=0\) if \(\phi_0 \neq 0\). His main result is as follows: Theorem 2.1: Let \(L=:L(A, \delta, \phi)\) be a generalized Cartan type \(H\) Lie algebra. Then \(H^2(L, F)\) is spanned by the cohomology classes \([\psi_1], [\psi_{2,i}]\), \([\psi_{3,i}]\) and \([\psi_{4,\mu}]\) for all \(i\in I\backslash \{0\}\) and all \(\mu \in M\).