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

Abstract

We generalize the linear algebra setting of Tate’s central extension to arbitrary dimension. In general, one obtains a Lie $\left(n+1\right)$-cocycle. We compute it to some extent. 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.