The Virasoro algebra

0 −→ c −→ g1 φ1 −→g −→ 0 such that c ⊆ Z(g1), the center of g̃. A morphism of central extensions is a Lie algebra homomorphism ψ : g1 → g2 such that φ2ψ = φ1. A universal central extension is a central extension g̃ such that there is a unique morphism from g̃ to every other central extension of g. The Schur multiplier is the kernel of the universal central extension of g. It classifies the projective representations of g. (at least this is right for GROUPS, see Steinberg). Isomorphism classes of one-dimensional central extensions are in bijection with elements c ∈ H2(g;F) via the formula [x, y]∼ = [x, y] + c(x, y)c, for x, y ∈ g,