The low-dimensional algebraic cohomology of the Witt and the Virasoro algebra

In this contribution, we provide the results on the low-dimensional algebraic cohomology with values in the trivial and the adjoint module of the Witt and the Virasoro algebra over a field K with char (K) = 0. A sketch of the various proofs is given, the details can be found in previous articles of the authors. More precisely, we give a summary of results known concerning the zeroth, first and second algebraic cohomology of the Witt and the Virasoro algebra, and we present some details about the more recent results related to the third algebraic cohomology. It has to be noted that we are dealing entirely with algebraic cohomology, meaning that our results are valid for any concrete realization of the Witt and the Virasoro algebra. Moreover, our results are independent of any underlying topology chosen.