On the nature of the Virasoro algebra

Abstract

The multiplication in the Virasoro algebra \[ [e_p, e_q] = (p - q) e_{p+q} + θ\left(p^3 - p\right) δ_{p + q}, \qquad p, q \in {\mathbf Z}, \] \[ [θ, e_p] = 0, \] 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 + εq) \over 1 + ε(p + q)} e_{p+q} + {1 \over 2} θ\left[p^3 - p + \left(ε- ε^{-1} \right) p^2 \right] δ^0_{p+q}, \vspace{3mm}\\ \ds e_p * θ= θ* 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.