2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/66800Using general principles of the theory of vertex operator algebras and their twisted modules, we obtain a bosonic, twisted construction of a certain central extension of a Lie algebra of differential operators on the circle, for an arbitrary twisting automorphism. The construction involves the Bernoulli polynomials in a fundamental way. This is explained through results in the general theory of vertex operator algebras, including a new identity, which we call ``modified weak associativity.'' This paper is an announcement. The detailed proofs will appear elsewhere.15 pages, LaTeX, Revised version (to appear in I.M.R.N.)Quantum AlgebraHigh Energy Physics - TheoryNumber TheoryRepresentation Theorytext