Lifting Lie algebras over the residue field of a discrete valuation ring (Q1124973)

Let \(R\) be a discrete valuation ring with maximal ideal \(\pi R\) and residue field \(k\), \({\mathcal L}\) a Lie algebra over \(R/\pi_kR\) and \(\overline{\mathcal L}= {\mathcal L}\otimes_Rk\). The authors show that there exists a class in \(H^3(\overline{\mathcal L},ad)\) such that \({\mathcal L}\) lifts to one over \(R/\pi^{k+1}R\) if and only if this class vanishes. Distinct lifts are in one to one correspondence with \(H^2(\overline{\mathcal L},ad)\).