The path group construction of Lie group extensions

We present an explicit realization of abelian extensions of infinite dimensional Lie groups using abelian extensions of path groups, by generalizing Mickelsson’s approach to loop groups and the approach of Losev–Moore–Nekrasov–Shatashvili to current groups. We apply our method to coupled cocycles on current Lie algebras and to Lichnerowicz cocycles on the Lie algebra of divergence free vector fields.