Non-connected Lie groups, twisted equivariant bundles and coverings

• Tipo: Artigo em Revista Científica Internacional
• Data: 2023

Autores

Barajas, G (Autor) Outra; García-Prada, O (Autor) Outra; Gothen, PB (Autor) FCUP; Riera, IMi (Autor) Outra

Revista

Título: Geometriae Dedicata

Vol. 217

ISSN: 0046-5755

Editora: Springer Nature

Outras Informações

ID Authenticus: P-00X-RAA

DOI: 10.1007/s10711-022-00764-w

Abstract (EN): <jats:title>Abstract</jats:title><jats:p>Let <jats:inline-formula><jats:alternatives><jats:tex-math>$$\Gamma $$</jats:tex-math><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>¿</mml:mi> </mml:math></jats:alternatives></jats:inline-formula> be a finite group acting on a Lie group <jats:italic>G</jats:italic>. We consider a class of group extensions <jats:inline-formula><jats:alternatives><jats:tex-math>$$1 \rightarrow G \rightarrow \hat{G} \rightarrow \Gamma \rightarrow 1$$</jats:tex-math><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mn>1</mml:mn> <mml:mo>¿</mml:mo> <mml:mi>G</mml:mi> <mml:mo>¿</mml:mo> <mml:mover> <mml:mi>G</mml:mi> <mml:mo>^</mml:mo> </mml:mover> <mml:mo>¿</mml:mo> <mml:mi>¿</mml:mi> <mml:mo>¿</mml:mo> <mml:mn>1</mml:mn> </mml:mrow> </mml:math></jats:alternatives></jats:inline-formula> defined by this action and a 2-cocycle of <jats:inline-formula><jats:alternatives><jats:tex-math>$$\Gamma $$</jats:tex-math><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>¿</mml:mi> </mml:math></jats:alternatives></jats:inline-formula> with values in the centre of <jats:italic>G</jats:italic>. We establish and study a correspondence between <jats:inline-formula><jats:alternatives><jats:tex-math>$$\hat{G}$$</jats:tex-math><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mover> <mml:mi>G</mml:mi> <mml:mo>^</mml:mo> </mml:mover> </mml:math></jats:alternatives></jats:inline-formula>-bundles on a manifold and twisted <jats:inline-formula><jats:alternatives><jats:tex-math>$$\Gamma $$</jats:tex-math><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>¿</mml:mi> </mml:math></jats:alternatives></jats:inline-formula>-equivariant bundles with structure group <jats:italic>G</jats:italic> on a suitable Galois <jats:inline-formula><jats:alternatives><jats:tex-math>$$\Gamma $$</jats:tex-math><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>¿</mml:mi> </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 <jats:inline-formula><jats:alternatives><jats:tex-math>$$\hat{G}$$</jats:tex-math><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mover> <mml:mi>G</mml:mi> <mml:mo>^</mml:mo> </mml:mover> </mml:math></jats:alternatives></jats:inline-formula>, since such a group is always isomorphic to an extension as above, where <jats:italic>G</jats:italic> is the connected component of the identity and <jats:inline-formula><jats:alternatives><jats:tex-math>$$\Gamma $$</jats:tex-math><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>¿</mml:mi> </mml:math></jats:alternatives></jats:inline-formula> is the group of connected components of <jats:inline-formula><jats:alternatives><jats:tex-math>$$\hat{G}$$</jats:tex-math><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mover> <mml:mi>G</mml:mi> <mml:mo>^</mml:mo> </mml:mover> </mml:math></jats:alternatives></jats:inline-formula>. </jats:p>

Idioma: Inglês

Tipo (Avaliação Docente): Científica