Vanishing of one-dimensional \(L^2\)-cohomologies of loop groups. (English) Zbl 1236.58020
Let \(G\) be a compact simply connected Lie group, and \(L\) the set of continuous loops in \(G\) beginning and ending at the identity element.
The author proves that every closed 1-form on \(L\) is the differential of a measureable function, and that the Hodge-Kodaira Laplacian (constructed from the left invariant flat connection on \(G\)) is injective on 1-forms on \(L\).
The author proves that every closed 1-form on \(L\) is the differential of a measureable function, and that the Hodge-Kodaira Laplacian (constructed from the left invariant flat connection on \(G\)) is injective on 1-forms on \(L\).
Reviewer: Benjamin McKay (Cork)
Keywords:
loop group; \(L^{2}\)-cohomology; Hodge-Kodaira Laplacian; Kodaira theorem; vanishing theorem; rough path analysisReferences:
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