Standard cocycles: Variations on themes of C. Kassel’s and R. Wilson’s

Central extensions of Lie algebras can be understood and classified by means of 2-cocycles. The Lie algebras we are interested in are “twisted forms” (defined by Galois descent) of algebras of the form with split finite-dimensional simple over a base field of characteristic 0 and a commutative unita...

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Published inForum mathematicum Vol. 29; no. 6; pp. 1441 - 1461
Main Authors Pianzola, Arturo, Prelat, Daniel, Sepp, Claudia
Format Journal Article
LanguageEnglish
Published Berlin De Gruyter 01.11.2017
Walter de Gruyter GmbH
Subjects
17B01
17B67
22E65
Algebra
Central extensions of Lie algebras
Galois descent
Lie groups
multiloop algebras
standard cocycle
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Summary:Central extensions of Lie algebras can be understood and classified by means of 2-cocycles. The Lie algebras we are interested in are “twisted forms” (defined by Galois descent) of algebras of the form with split finite-dimensional simple over a base field of characteristic 0 and a commutative unital and associative -algebra (such algebras are ubiquitous in modern infinite-dimensional Lie theory). We introduce a special type of cocycle that we called . Our main result shows that any cocycle is cohomologous to a unique standard cocycle. As an application we give a precise description of the universal central extension of the twisted forms of mentioned above. This yields a new proof of a classic theorem of C. Kassel [ ]. For multiloop algebras, we obtain a “twisted” version of Kassel’s result (which is due to R. Wilson [ ] in the case of the affine Kac–Moody Lie algebras).
ISSN:0933-7741
1435-5337
DOI:10.1515/forum-2016-0148