Homomorphisms inducing isomorphisms in Lie algebra cohomology (Q1313814)

Let \(V(L)\) denote the associated restricted universal enveloping algebra of a restricted Lie algebra \(L\) over an algebraically closed field \(k\) of characteristic \(p>0\). If \(f: L\to L'\) is a map of restricted Lie algebras, the author investigates the relationship between \(f\) and the restriction map in cohomology \(\text{res}(f): H^* (V(L'), k)\to H^* (V(L), k)\) induced by \(f\). For instance, it is shown that if \(f\) is an epimorphism and \(\text{res}(f): H^{2^*} (V(L'), k)\to H^{2^*} (V(L), k)\) is an isomorphism, then \(\ker f\) is a torus.