The Vanishing of Low-Dimensional Cohomology Groups of the Witt and the Virasoro algebra

A proof of the vanishing of the first and the third cohomology groups of the Witt algebra with values in the adjoint module is given. The proofs given in the present article are completely algebraic and independent of any underlying topology. They are a generalization of the ones provided by Schlichenmaier, who proved the vanishing of the second cohomology group using purely algebraic methods. In the case of the third cohomology group though, extra difficulties arise and the involved proofs are distinctly more complicated.