Homotopy commutativity in Hermitian symmetric spaces

Ganea proved that the loop space of $\mathbb{C} P^n$ is homotopy commutative if and only if $n=3$ . We generalize this result to that the loop spaces of all irreducible Hermitian symmetric spaces but $\mathbb{C} P^3$ are not homotopy commutative. The computation also applies to determining the homot...

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Bibliographic Details
Published in:Glasgow mathematical journal 2022-09, Vol.64 (3), p.746-752
Main Authors: Kishimoto, Daisuke, Takeda, Masahiro, Tong, Yichen
Format: Article
Language:English
Subjects:
Commutativity
Lie groups
Toruses
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Summary:Ganea proved that the loop space of $\mathbb{C} P^n$ is homotopy commutative if and only if $n=3$ . We generalize this result to that the loop spaces of all irreducible Hermitian symmetric spaces but $\mathbb{C} P^3$ are not homotopy commutative. The computation also applies to determining the homotopy nilpotency class of the loop spaces of generalized flag manifolds $G/T$ for a maximal torus T of a compact, connected Lie group G.
ISSN:0017-0895
1469-509X
DOI:10.1017/S0017089522000118