The multiplication in the Virasoro algebra \[ [e_p, e_q] = (p - q) e_{ p+q} + \theta \left(p^3 - p\right) \delta_{ p + q}, \qquad p, q \in { \mathbf Z}, \] \[ [\theta, e_p] = 0, \] 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}{ $*$} \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 ${ \cal R}$ satisfies the property \renewcommand{ \theequation}{ $**$} \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}$({ \cal R})$ 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 ${ \cal R}$ becomes associative. Formula $(*)$ above also has a differential-variational counterpart.
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