The multiplication in the Virasoro algebra
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\begin{document}$$\begin{gathered} {[{e_p},{e_q}]} = (p - q){e_{p + q}} + \theta \,({p^3} - p)\,{\delta _{p + q}},\quad p,q \in \,Z, \\ [\theta ,{e_p}] = 0,\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\; \hfill \\ \end{gathered} $$\end{document}
comes from the commutator [ep, eq] = ep * eq – eq * ep in a quasiassociative algebra with the multiplication
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\begin{document}$$\begin{gathered} {e_p}*{e_q} = - \frac{{q(1 +\epsilon q)}}{{1 + \epsilon(p + q)}}{e_{p + q}} + \frac{1}{2}\theta [{p^3} - p + (\epsilon - \epsilon{^{ - 1}}){p^2}]\,\delta _{p + q}^0, \hfill \\ {e_p}*\theta = \,\theta *{e_p} = 0.\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\; \hfill \\ \end{gathered}$$\end{document}
The multiplication in a quasiassociative algebra R satisfies the property
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\begin{document}$$a*(b*c) - (a*b)*c = b*(a*c) - (b*a)*c,\quad a,b,c \in \mathcal{R}.$$\end{document}
This propertyis necessaryand sufficient for the Lie algebra Lie(\documentclass[12pt]{minimal}
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\begin{document}$$\mathcal{R}$$\end{document}) 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 \documentclass[12pt]{minimal}
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\begin{document}$$\mathcal{R}$$\end{document} becomes associative. Formula (*) above also has a differential-variational counterpart.
