#### Vol. 8, No. 1, 2014

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

### Oliver Braunling

Vol. 8 (2014), No. 1, 19–52
##### 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.

##### Keywords
adèle, residue symbol, Tate central extension, Kac–Moody, Japanese group
##### Mathematical Subject Classification 2010
Primary: 17B56, 17B67
Secondary: 32A27