The study of cohomology of infinite-dimensional Lie algebras was started by Gel'fand and Fuchs in the late 1960s. Since then, significant progress has been made, mainly focusing on the Witt algebra (the Lie algebra of vector fields on the punctured affine line) and some of its subalgebras. In this talk, I will explain the basics of Lie algebra cohomology and sketch the computation of the first cohomology group of certain subalgebras of the Witt algebra known as submodule-subalgebras. Interestingly, these cohomology groups are, in some sense, controlled by the cohomology of the Witt algebra. This can be explained by the fact that the Witt algebra can be abstractly reconstructed from any of its submodule-subalgebras, which can be described as a universal property satisfied by the Witt algebra.
