Isometric Lie 2-group actions on Riemannian groupoids

We study isometric actions of Lie 2-groups on Riemannian groupoids by exhibiting some of their immediate properties and implications. Firstly, we prove an existence result which allows both to obtain 2-equivariant versions of the Slice Theorem and the Equivariant Tubular Neighborhood Theorem and to construct bi-invariant groupoid metrics on compact Lie 2-groups. We provide natural examples, transfer some classical constructions and explain how this notion of isometric 2-action yields a way to develop a 2-equivariant Morse theory on Lie groupoids. Secondly, we give an infinitesimal description of an isometric Lie 2-group action. We define an algebra of transversal infinitesimal isometries associated to any Riemannian n-metric on a Lie groupoid which in turn gives rise to a notion of geometric Killing vector field on a quotient Riemannian stack. If our Riemannian stack is separated then we prove that the algebra formed by such geometric Killing vector fields is always finite dimensional.

• Autores: Valencia Quintero, Fabricio; Herrera Carmona, Juan Sebastián
• Tipo de recurso: Article of investigation
• Fecha de publicación: 2023
• Institución: Universidad de Antioquia
• Repositorio: Repositorio UdeA
• Idioma: eng
• OAI Identifier: oai:bibliotecadigital.udea.edu.co:10495/46232
• Acceso en línea: https://hdl.handle.net/10495/46232
• Palabra clave: Grupos de Lie; Lie groups; Variedades de Riemann; Groupoides; Groupoids; Isometría (Matemáticas); Isometrics (Mathematics); Álgebra vectorial; Vector algebra; Equivariant Tubular Neighborhood Theorem; Teorema de Vecindad Tubular Equivariante; Slice Theorem; Equivariant Tubular Neighborhood Theorem
• Rights: openAccess
• License: http://creativecommons.org/licenses/by-nc-sa/4.0/