Standard cocycles: Variations on themes of C. Kassel’s and R. Wilson’s
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Bibliographic record
Abstract
Abstract 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 <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mi>𝔤</m:mi> <m:msub> <m:mo>⊗</m:mo> <m:mi>k</m:mi> </m:msub> <m:mi>R</m:mi> </m:mrow> </m:math> {{\mathfrak{g}}\otimes_{k}R} with <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mi>𝔤</m:mi> </m:math> {{\mathfrak{g}}} split finite-dimensional simple over a base field k of characteristic 0 and R a commutative unital and associative k -algebra (such algebras are ubiquitous in modern infinite-dimensional Lie theory). We introduce a special type of cocycle that we called standard . 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 <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mi>𝔤</m:mi> <m:msub> <m:mo>⊗</m:mo> <m:mi>k</m:mi> </m:msub> <m:mi>R</m:mi> </m:mrow> </m:math> {{\mathfrak{g}}\otimes_{k}R} mentioned above. This yields a new proof of a classic theorem of C. Kassel [8]. For multiloop algebras, we obtain a “twisted” version of Kassel’s result (which is due to R. Wilson [21] in the case of the affine Kac–Moody Lie algebras).
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Codex and Gemma teacher scores by category
| Category | Codex | Gemma |
|---|---|---|
| Metaresearch | 0.000 | 0.000 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.000 | 0.000 |
| Bibliometrics | 0.000 | 0.000 |
| Science and technology studies | 0.000 | 0.000 |
| Scholarly communication | 0.000 | 0.000 |
| Open science | 0.000 | 0.000 |
| Research integrity | 0.000 | 0.000 |
| Insufficient payload (model declined to judge) | 0.000 | 0.000 |
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