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.

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.

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## Additional indexing

Item Type: | Dissertation |
---|---|

Referees: | Cattaneo A S, Felder G |

Communities & Collections: | 07 Faculty of Science > Institute of Mathematics |

Dewey Decimal Classification: | 510 Mathematics |

Language: | English |

Date: | 2008 |

Deposited On: | 07 Jan 2009 14:25 |

Last Modified: | 05 Apr 2016 12:44 |

Number of Pages: | 137 |

Related URLs: | http://opac.nebis.ch/F/?local_base=NEBIS&con_lng=GER&func=find-b&find_code=SYS&request=005670322 |

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