On formal series and infinite products over Lie algebras

A brief survey of new methods for the study of nonstandard associative envelopes of Lie algebras is presented. Various extensions of the universal enveloping algebra $U\mathfrak g$ are considered, where $\mathfrak g$ is a symmetrizable Kac–Moody algebra. An elementary proof is given for describing the “extremal projector” over $\mathfrak g$ as an infinite product over $U\mathfrak g$. Certain applications to the theory of $\mathfrak g$-modules are discussed.