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Universal central extensions of gauge algebras and groups
 J. Reine Angew. Math
"... Abstract We show that the canonical central extension of the group of sections of a Lie group bundle over a compact manifold, constructed in [NW09], is universal. In doing so, we prove universality of the corresponding central extension of Lie algebras in a slightly more general setting. I Setting ..."
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Abstract We show that the canonical central extension of the group of sections of a Lie group bundle over a compact manifold, constructed in [NW09], is universal. In doing so, we prove universality of the corresponding central extension of Lie algebras in a slightly more general setting. I Setting of the Problem Let K → M be a finitedimensional, locally trivial bundle of Lie algebras. A cocycle on its Lie algebra of sections can then be constructed as follows. For any Lie algebra k, the derivations der(k) act naturally on its second symmetric tensor power , and we denote the quotient by satisfies κ(d(x), y) + κ(x, d(y)) = 0 for all d ∈ der(k) and is universal with this property. Now let ∇ be a Lie connection on K, i.e. one that satisfies where V (K) is the vector bundle one obtains by applying k → V (k) fibrewise. The connection d ∇ does not depend on ∇, as any two Lie connections differ by a pointwise derivation, which acts trivially on V (K). We therefore omit the subscript and simply write d. Using the identity d κ(ξ, η) = κ(∇ξ, η) + κ(ξ, ∇η) and the compatibility of ∇ with the Lie bracket, it is not hard to check that ω ∇