On the nature of the Virasoro algebra

The multiplication in the Virasoro algebra $$[e_p,e_q]=(p-q)e_{p+q}+\theta (p^3-p){\delta }_{p+q},p,q\in \mathbf{Z},$$ $$[\theta ,e_p]=0,$$ comes from the commutator $[e_p, e_q] = e_p * e_q - e_q * e_p$ in a quasiassociative algebra with the multiplication \renewcommand{\theequation}{$\ast$} \be \ba{l} \ds e_p * e_q = - {q (1 + \epsilon q) \over 1 + \epsilon (p + q)} e_{p+q} + {1 \over 2} \theta \left[p^3 - p + \left(\epsilon - \epsilon^{-1} \right) p^2 \right] \delta^0_{p+q}, \vspace{3mm}\ \ds e_p * \theta = \theta* e_p = 0. \ea \ee The multiplication in a quasiassociative algebra $\mathcal{R}$ satisfies the property \renewcommand{\theequation}{$\ast \ast$} \be a * (b * c) - (a * b) * c = b * (a * c) - (b * a) * c, \qquad a, b, c \in {\cal R}. \ee This property is necessary and sufficient for the Lie algebra {\it Lie}$(\mathcal{R})$ 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 $\mathcal{R}$ becomes associative. Formula $(\ast )$ above also has a differential-variational counterpart.