Algebra & Number Theory

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

Oliver Braunling

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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 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.

Article information

Algebra Number Theory, Volume 8, Number 1 (2014), 19-52.

Received: 16 June 2012
Revised: 14 April 2013
Accepted: 9 September 2013
First available in Project Euclid: 20 December 2017

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 17B56: Cohomology of Lie (super)algebras 17B67: Kac-Moody (super)algebras; extended affine Lie algebras; toroidal Lie algebras
Secondary: 32A27: Local theory of residues [See also 32C30]

adèle residue symbol Tate central extension Kac–Moody Japanese group


Braunling, Oliver. Adèle residue symbol and Tate's central extension for multiloop Lie algebras. Algebra Number Theory 8 (2014), no. 1, 19--52. doi:10.2140/ant.2014.8.19.

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