Standard cocycles: variations on themes of C. Kassel’s and R. Wilson’s. (English) Zbl 1432.17017
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 \(\mathfrak{g}\otimes_k R\) with \(\mathfrak{g}\) 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 \(\mathfrak{g}\otimes_k R\) mentioned above. This yields a new proof of a classic theorem of C. Kassel [J. Pure Appl. Algebra 34, 265–275 (1984; Zbl 0549.17009)]. For multiloop algebras, we obtain a “twisted” version of Kassel’s result (which is due to R. L. Wilson [Lie algebras and related topics, Lect. Notes Math. 933, 210–213 (1982; Zbl 0498.17012)] in the case of the affine Kac-Moody Lie algebras).
MSC:
| 17B55 | Homological methods in Lie (super)algebras |
| 17B67 | Kac-Moody (super)algebras; extended affine Lie algebras; toroidal Lie algebras |
| 17B01 | Identities, free Lie (super)algebras |
| 22E65 | Infinite-dimensional Lie groups and their Lie algebras: general properties |
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