On the nature of the Virasoro algebra

The multiplication in the Virasoro algebra [e(p), e(q)] = (p - q)e(p + q) + theta (p(3) - p) delta(p+q), p, q is an element of Z, [theta, e(p)] = 0, comes from the commutator [e(p), e(q)] = e(p) * e(q) - e(p) in a quasiassociative algebra with the multiplication e(p) * e(q) = - q(1+epsilon q)/1+epsilon(p+q) e(p+q) + 1/2 theta [p(3) - p + (epsilon - epsilon(-1))p(2)] delta(p+q)(0), (*) e(p) * theta = theta * e(p) = 0. 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 is an element of 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 quasiassociative algebra R becomes associative. Formula (*) above also has a differential-variational counterpart.