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We consider a class of group extensions $$1 \\rightarrow G \\rightarrow \\hat{G} \\rightarrow \\Gamma \\rightarrow 1$$<\/jats:tex-math>\n \n 1<\/mml:mn>\n \u2192<\/mml:mo>\n G<\/mml:mi>\n \u2192<\/mml:mo>\n \n G<\/mml:mi>\n ^<\/mml:mo>\n <\/mml:mover>\n \u2192<\/mml:mo>\n \u0393<\/mml:mi>\n \u2192<\/mml:mo>\n 1<\/mml:mn>\n <\/mml:mrow>\n <\/mml:math><\/jats:alternatives><\/jats:inline-formula> defined by this action and a 2-cocycle of $$\\Gamma $$<\/jats:tex-math>\n \u0393<\/mml:mi>\n <\/mml:math><\/jats:alternatives><\/jats:inline-formula> with values in the centre of G<\/jats:italic>. We establish and study a correspondence between $$\\hat{G}$$<\/jats:tex-math>\n \n G<\/mml:mi>\n ^<\/mml:mo>\n <\/mml:mover>\n <\/mml:math><\/jats:alternatives><\/jats:inline-formula>-bundles on a manifold and twisted $$\\Gamma $$<\/jats:tex-math>\n \u0393<\/mml:mi>\n <\/mml:math><\/jats:alternatives><\/jats:inline-formula>-equivariant bundles with structure group G<\/jats:italic> on a suitable Galois $$\\Gamma $$<\/jats:tex-math>\n \u0393<\/mml:mi>\n <\/mml:math><\/jats:alternatives><\/jats:inline-formula>-covering of the manifold. We also describe this correspondence in terms of non-abelian cohomology. Our results apply, in particular, to the case of a compact or reductive complex Lie group $$\\hat{G}$$<\/jats:tex-math>\n \n G<\/mml:mi>\n ^<\/mml:mo>\n <\/mml:mover>\n <\/mml:math><\/jats:alternatives><\/jats:inline-formula>, since such a group is always isomorphic to an extension as above, where G<\/jats:italic> is the connected component of the identity and $$\\Gamma $$<\/jats:tex-math>\n \u0393<\/mml:mi>\n <\/mml:math><\/jats:alternatives><\/jats:inline-formula> is the group of connected components of $$\\hat{G}$$<\/jats:tex-math>\n \n G<\/mml:mi>\n ^<\/mml:mo>\n <\/mml:mover>\n <\/mml:math><\/jats:alternatives><\/jats:inline-formula>.\n<\/jats:p>","DOI":"10.1007\/s10711-022-00764-w","type":"journal-article","created":{"date-parts":[[2023,1,18]],"date-time":"2023-01-18T06:03:13Z","timestamp":1674021793000},"update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":3,"title":["Non-connected Lie groups, twisted equivariant bundles and coverings"],"prefix":"10.1007","volume":"217","author":[{"given":"G.","family":"Barajas","sequence":"first","affiliation":[]},{"given":"O.","family":"Garc\u00eda-Prada","sequence":"additional","affiliation":[]},{"given":"P. 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