2 Virasoro as a local Lie algebra on C 3 2.1 Dolbeault resolution of holomorphic vector fields . . . . . . . . . . . . . . . . . . . . . . . . . 3 2.2 Lie algebra extensions and the cocycle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 2.2.1 Extensions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 2.2.2 A cocycle for LC . . . . . ....

The multiplication in a quasiassociative algebra R satisfies the property a ∗ (b ∗ c)− (a ∗ b) ∗ c = b ∗ (a ∗ c)− (b ∗ a) ∗ c, a, b, c ∈ R. (∗∗) This property is necessary and sufficient for the Lie algebra Lie(R) to have a phase space. The above formulae are put into a cohomological framework, with the relevant complex being different from the Hochschild one even when the relevant quasiassocia...