Cohomology of nilpotent subalgebras of the algebra \(\widehat{sl_2}\) (Q1364879)

The algebra of polynomial \(sl_2\)-currents on the line is considered, and the nilpotent subalgebras of the type \[ N(P_+, P_0, P_-):= P_+(t)\mathbb{C} [t]\otimes e+P_0(t) \mathbb{C}[t]\otimes h+P_-(f)\cdot \mathbb{C}[t]\otimes f \] are the main object of study, where \(P_0, P_+, P_-\) are polynomials such that \(P_0\) divides \(P_+\cdot P_-\) and \(e,h,f\) are the standard generators of \(sl_2\). The general result, Theorem 1, is if \(\text{deg }P_++ \text{deg }P_->0\) then the dimension of the homology \(H_i(N(P_+, P_0, P_-))\) depends only on the degrees of \(P_0, P_+, P_-\). Actually the dimensions are computed. The next two theorems deal with specific polynomials. In Theorem 3 the bigraded space structure in \(H_i(N (t^{h_+}, t^{h_0}, t^{h_-}))\) is computed, and in Theorem 2 the case when \(P_0, P_+, P_-\) has no multiple roots is worked out in more detail. The authors point out the difficulty they have met when trying to replace \(sl_2\) by another semisimple Lie algebra.