Cohomologies of infinite-dimensional Borel algebras (Q2639158)

The cohomologies \(H^*(B,B)\) are studied, where B is a Borel subalgebra of the affine Lie algebra \(G=\oplus_{k\in {\mathbb{Z}}}G_ k\). Using the Hochschild-Serre spectral sequence the spectral sequence \(E_ 2^{ij}=C^ i(H,H^ j(N,N)^{G_ 0})\) converging to \(H^*(B,B)\), \(N=\oplus_{k>0}G_ k\), is constructed. For the non-twisted algebra G it is proved that \(H^ n(B,B)=0\) for \(n\leq 2\).