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

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