# Weil Reciprocity Law and the Theorem of Residues

## TL;DR

This paper demonstrates how the Theorem of Residues and the Gelfand-Fuchs cocycle can be derived from the Weil Reciprocity Law, linking group extensions to Lie algebra statements.

## Contribution

It provides a unified and simplified derivation of key theorems in complex analysis and Lie algebra cohomology from the Weil Reciprocity Law.

## Key findings

- TR and Gelfand-Fuchs cocycle derived from WRL
- WRL viewed as triviality of certain group extensions
- Lie algebra version of TR established

## Abstract

This paper shows how the Theorem of Residues (TR) and the Gelfand-Fuchs cocycle can be deduced in a simple way from the Weil Reciprocity Law (WRL). Indeed, if one understand WRL as the triviality of certain extension of groups, then TR is the same statement at the level of Lie algebras. Finally, the Gelfand-Fuchs cocycle can also be obtained in this way.