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Let $G$ be a Lie group with Lie algebra $g$. As it is well known the Maurer-Cartan form $ω:TG\rightarrow g$ transports any vector $X\in T_{x}G$ to the start $l_{x^{-1}*}(X)\in g$, $l_{x^{-1}}$ denoting the left translation. Let $σ:[0,1]\rightarrow G$ a smooth path on $G$. It there a way to define path integration on $G$ such that $\int_{σ}{ω}=σ(1)σ(0)^{-1}$?

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