Vanishing of one dimensional L^2-cohomologies of loop groups

Abstract

Let $G$ be a simply connected compact Lie group. Let $L_e(G)$ be the based loop group with the base point $e$ which is the identity element. Let ${\nu }_e$ be the pinned Brownian motion measure on $L_e(G)$ and let $\alpha \in L^2({\wedge }^1{T}^{\ast }{L}_e(G),{\nu }_e)\cap {\mathbb{D}}^{\infty ,p}({\wedge }^1{T}^{\ast }{L}_e(G),{\nu }_e)$ $(1<p<2)$ be a closed 1-form on $L_e(G)$. Using results in rough path analysis, we prove that there exists a measurable function $f$ on $L_e(G)$ such that $df=\alpha$. Moreover we prove that $\dim \ker \square =0$ for the Hodge-Kodaira type operator $\square$ acting on 1-forms on $L_e(G)$.

References in corpus (3)

• Dirichlet spaces on H-convex sets in Wiener space
• An L2 theory for differential forms on path spaces I
• Weak Poincare inequalities on domains defined by Brownian rough paths

Cited by in corpus (9)

• Reconstruction for the Signature of a Rough Path
• The vanishing of L2 harmonic one-forms on based path spaces
• Large deviation principle of Freidlin-Wentzell type for pinned diffusion processes
• Short time full asymptotic expansion of hypoelliptic heat kernel at the cut locus
• Large deviations for rough path lifts of Watanabe's pullbacks of delta functions
• Quasi-sure Existence of Gaussian Rough Paths and Large Deviation Principles for Capacities
• Asymptotics of spectral gaps on loop spaces over a class of Riemannian manifolds
• Semi-classical limit of the generalized second lowest eigenvalue of Dirichlet Laplacians on small domains in path spaces
• Large deviations for small noise hypoelliptic diffusion bridges on sub-Riemannian manifolds