Cohomology of nilradicals of Borel subalgebras

Abstract:

Let $\mathfrak {N}$ be the maximal nilpotent ideal in a Borel subalgebra of a complex simple Lie algebra. The cohomology groups ${H^1}(\mathfrak {N},\mathfrak {N}),{H^1}(\mathfrak {N},{\mathfrak {N}^\ast })$ and the $\mathfrak {N}$-invariant symmetric bilinear forms on $\mathfrak {N}$ are determined. The main result is the computation of ${H^2}(\mathfrak {N},\mathfrak {N})$.