The low-dimensional algebraic cohomology of infinite-dimensional Lie algebras of Virasoro-type

In this doctoral thesis, the low-dimensional algebraic cohomology of infinite-dimensional Lie algebras of Virasoro-type is investigated. The considered Lie algebras include the Witt algebra, the Virasoro algebra and the multipoint Krichever-Novikov vector field algebra. We consider algebraic cohomology, meaning we do not put any constraints of continuity on the cochains. The Lie algebras are considered as abstract Lie algebras in the sense that we do not work with particular realizations of the Lie algebras. The results are thus independent of any underlying choice of topology.
The thesis is self-contained, as it starts with a technical chapter introducing the definitions, concepts and methods that are used in the thesis. For motivational purposes, some time is spent on the interpretation of the low-dimensional cohomology. First results include the computation of the first and the third algebraic cohomology of the Witt and the Virasoro algebra with values in the trivial and the adjoint module, the second algebraic cohomology being known already. A canonical link between the low-dimensional cohomology of the Witt and the Virasoro algebra is exhibited by using the Hochschild-Serre spectral sequence. More results are given by the computation of the low-dimensional algebraic cohomology of the Witt and the Virasoro algebra with values in general tensor-densities modules. The study consists of a mix between elementary algebra and algorithmic analysis. Finally, some results concerning the low-dimensional algebraic cohomology of the multipoint Krichever-Novikov vector field algebra are derived. The thesis is concluded with an outlook containing possible short-term goals that could be achieved in the near future as well as some long-term goals.