Formal and analytic deformations of the Witt algebra
The Witt algebra \(W\) (or the Virasoro algebra without central term) is the complex Lie algebra with basis \(\{l_n\mid n\in{\mathbb Z} \}\) fulfilling the commutation relations \([l_n,l_m]=(m-n)l_{n+m}\). In one of its standard realisations: \(l_n=z^{n+1}d/dz\). In this article formal deformations, i.e. Lie algebra structures on \(Wt\) and analytic deformations of an adic completion of the Witt algebra \(\widehat{W}\) (i.e. passing from Laurent polynomials to Laurent series) are considered. More precisely, certain deformations \(W_{tf}\), resp. \(\widehat{W}_{tf}\) defined over the space of even polynomials \(f\) in the variable \(z\) are studied. From cohomological arguments it follows \(Wt\cong W_{tf}\) over \(\mathbb Ct\). The author shows that one also obtains \(\widehat{W}\cong \widehat{W}_{tf}\) for the adic completion . This is extended to the smooth situation (the space of smooth vector fields on the sphere \(S^1\)) \(\widehat{W}^{\infty}\cong \widehat{W}_{f}^{\infty}\) if \(f\) is close to 0. In this case certain results on the holomorphic dependence on \(f\) are obtained. As special examples of this type of deformation, the Krichever-Novikov algebras on the torus studied by \textit{A. Ruffing, T. Deck} and \textit{M. Schlichenmaier} [Lett. Math. Phys. 26, 23-32 (1992; Zbl 0774.17027)] are considered. In this context it should be noted that the adically completed Krichever-Novikov algebra with respect to the Krichever-Novikov basis is isomorphic to the adically completed Witt algebra with respect to an arbitrary choice of local coordinates.
scientific article; zbMATH DE number 1269745