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I have a question about Section 8.6 of Pressley-Segal's Loop groups book. Let $G$ be a compact, connected Lie group. Proposition 8.6.6 concerns the comparison of homotopy type between its polynomial loop group $L_{pol}G$ (c.f. Section 3.5 of the book) and $LG$, the smooth loop group.

I cannot tell if this result requires/assumes semi-simplicity or not? The authors start the section by discussing semi-simplicity but also refer to a "general compact group" $G$ just before the theorem (and in the title of the section). Also, when it is quoted in the literature, some people assume semi-simplicity and others don't.

Semi-simplicity is clearly not a necessary assumption, for example if you check the discussion just above Prop 3.5.3 of Pressley-Segal, we see that it's fine for a torus. They also do the case of $U(n)$ in Section 8.4; see also Pressley's earlier 1980 Topology paper, Decompositions of the space of loops on a lie group, https://doi.org/10.1016/0040-9383(80)90032-4

I could imagine a simple argument that combines the torus case and the semi-simple case (using maybe the fact that up to a finite cover, $G$ is diffeomorphic to a product of these), but I'm not competent in Lie groups and don't feel that confident.

Can anyone clarify what the minimal hypotheses should be?

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  • $\begingroup$ Admitting a faithful finite-dimensional representation? Not sure, but it seems close... $\endgroup$
    – David Roberts
    Commented Feb 9, 2022 at 8:50
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    $\begingroup$ @DavidRoberts For my purposes the compact case is enough. $\endgroup$ Commented Feb 9, 2022 at 15:15

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