Hochschild-Serre Spectral Sequence for Lie Conformal Algebras

Abstract

Lie conformal algebras, originally introduced by Kac, encode an axiomatic description of the operator product expansion of chiral fields in conformal field theory. In particular, Lie conformal algebras provide a powerful tool for studying the infinite-dimensional Lie algebras and associative algebras satisfying the locality property. The most important examples of Lie conformal algebra include the Virasoro algebra Vir, the current algebra Cur g, and their semidirect product of Vir and Cur g. In this thesis, we construct the Hochschild–Serre spectral sequence for Lie conformal algebras. This construction follows a similar approach to the original work done by G. Hochschild and J-P. Serre for the case of Lie algebras. In addition, we describe the inflation-restriction exact sequence, a special case of the five-term exact sequence. As an application of this construction, we provide a different approach for calculating the cohomology of the semidirect product Vir and Cur g with trivial coefficients. In addition, we offer explicit computations for the basic cohomology of Vir and the semidirect product of Vir and Cur g with coefficients in their finite conformal irreducible modules.