# On morphic actions and integrability of LA-groupoids

Stefanini, L. On morphic actions and integrability of LA-groupoids. 2008, University of Zurich, Faculty of Science.

## Abstract

Lie theory for the integration of Lie algebroids to Lie groupoids, on the one hand, and
of Poisson manifolds to symplectic groupoids, on the other, has undergone tremendous
developements in the last decade, thanks to the work of Mackenzie-Xu, Moerdijk-Mrˇcun,
Cattaneo-Felder and Crainic-Fernandes, among others.
In this thesis we study - part of - the categorified version of this story, namely the
integrability of LA-groupoids (groupoid objects in the category of Lie algebroids), to
double Lie groupoids (groupoid objects in the category of Lie groupoids) providing a
first set of sufficient conditions for the integration to be possible.
Mackenzie’s double Lie structures arise naturally from lifting processes, such as the
cotangent lift or the path prolongation, on ordinary Lie theoretic and Poisson geometric
objects and we use them to study the integrability of quotient Poisson bivector fields,
the relation between “local” and “global” duality of Poisson groupoids and Lie theory
for Lie bialgebroids and Poisson groupoids.
In the first Chapter we prove suitable versions of Lie’s 1-st and 2-nd theorem for Lie
bialgebroids, that is, the integrability of subobjects (coisotropic subalgebroids) and morphisms,
extending earlier results by Cattaneo and Xu, obtained using different techniques.
We develop our functorial approach to the integration of LA-groupoids in the second
Chapter, where we also obtain partial results, within the program, proposed by Weinstein,
for the integration of Poisson groupoids to symplectic double groupoids.
The task of integrating quotients of Poisson manifolds with respect to Poisson groupoid
actions motivates the study we undertake in third Chapter of what we refer to as morphic
actions, i.e. groupoid actions in the categories of Lie algebroids and Lie groupoids, where
we obtain general reduction and integrability results.
In fact, applying suitable procedures a la Marsden-Weinstein zero level reduction to
“moment morphisms”, respectively of Lie bialgebroids or Poisson groupoids, canonically
associated to a Poisson G-space, we derive two approches to the integration of the quotient
Poisson bivector fields.
The first, a kind of integration via symplectic double groupoids, is not always effective
but reproduces the “symplectization functor” approch to Poisson actions of Lie groups,
very recently developed by Fernandes-Ortega-Ratiu, from quite a different perspective.
We earlier implemented this approach successfully in the special case of complete Poisson
groups.
The second approach, relying both on a cotangent lift of the Poisson G-space and on a
prolongation of the original action to an action on suitable spaces of Lie algebroid homotopies,
produces necessary and sufficient integrability conditions for the integration and
gives a positive answer to the integrability problem under the most natural assumptions.

Lie theory for the integration of Lie algebroids to Lie groupoids, on the one hand, and
of Poisson manifolds to symplectic groupoids, on the other, has undergone tremendous
developements in the last decade, thanks to the work of Mackenzie-Xu, Moerdijk-Mrˇcun,
Cattaneo-Felder and Crainic-Fernandes, among others.
In this thesis we study - part of - the categorified version of this story, namely the
integrability of LA-groupoids (groupoid objects in the category of Lie algebroids), to
double Lie groupoids (groupoid objects in the category of Lie groupoids) providing a
first set of sufficient conditions for the integration to be possible.
Mackenzie’s double Lie structures arise naturally from lifting processes, such as the
cotangent lift or the path prolongation, on ordinary Lie theoretic and Poisson geometric
objects and we use them to study the integrability of quotient Poisson bivector fields,
the relation between “local” and “global” duality of Poisson groupoids and Lie theory
for Lie bialgebroids and Poisson groupoids.
In the first Chapter we prove suitable versions of Lie’s 1-st and 2-nd theorem for Lie
bialgebroids, that is, the integrability of subobjects (coisotropic subalgebroids) and morphisms,
extending earlier results by Cattaneo and Xu, obtained using different techniques.
We develop our functorial approach to the integration of LA-groupoids in the second
Chapter, where we also obtain partial results, within the program, proposed by Weinstein,
for the integration of Poisson groupoids to symplectic double groupoids.
The task of integrating quotients of Poisson manifolds with respect to Poisson groupoid
actions motivates the study we undertake in third Chapter of what we refer to as morphic
actions, i.e. groupoid actions in the categories of Lie algebroids and Lie groupoids, where
we obtain general reduction and integrability results.
In fact, applying suitable procedures a la Marsden-Weinstein zero level reduction to
“moment morphisms”, respectively of Lie bialgebroids or Poisson groupoids, canonically
associated to a Poisson G-space, we derive two approches to the integration of the quotient
Poisson bivector fields.
The first, a kind of integration via symplectic double groupoids, is not always effective
but reproduces the “symplectization functor” approch to Poisson actions of Lie groups,
very recently developed by Fernandes-Ortega-Ratiu, from quite a different perspective.
We earlier implemented this approach successfully in the special case of complete Poisson
groups.
The second approach, relying both on a cotangent lift of the Poisson G-space and on a
prolongation of the original action to an action on suitable spaces of Lie algebroid homotopies,
produces necessary and sufficient integrability conditions for the integration and
gives a positive answer to the integrability problem under the most natural assumptions.

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Item Type: Dissertation Cattaneo A S, Felder G 07 Faculty of Science > Institute of Mathematics 510 Mathematics English 2008 07 Jan 2009 14:25 05 Apr 2016 12:44 137 http://opac.nebis.ch/F/?local_base=NEBIS&con_lng=GER&func=find-b&find_code=SYS&request=005670322
Permanent URL: https://doi.org/10.5167/uzh-8665