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

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 𝔤 ⊗ k R {{\mathfrak{g}}\otimes_{k}R} with 𝔤 {{\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 𝔤 ⊗ k R {{\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).