Adele residue symbol and Tate's central extension for multiloop Lie algebras

TL;DR

This paper extends Tate's central extension and residue symbol concepts to higher dimensions, deriving explicit formulas for Lie (n+1)-cocycles using a multidimensional adelic residue approach.

Contribution

It introduces a higher-dimensional generalization of Tate's residue symbol and central extension, providing explicit cocycle computations for multiloop Lie algebras.

Findings

Explicit formula for Lie (n+1)-cocycle in higher dimensions

Generalization of Tate's residue symbol to multidimensional setting

Connection between adelic residues and Lie algebra extensions

Abstract

We generalize the linear algebra setting of Tate's central extension to arbitrary dimension. In general, one obtains a Lie (n+1)-cocycle. We compute it explicitly. The construction is based on a Lie algebra variant of Beilinson's adelic multidimensional residue symbol, generalizing Tate's approach to the local residue symbol for 1-forms on curves.