Lie groupoid and Lie algebroid week in Coimbra IV

Meeting on modern aspects of the geometry and applications of Lie groupoids and Lie algebroids

This meeting will contain 2 mini-courses and various talks, aiming to introduce some recent advances of the theory of Lie groupoids, Lie algebroids, and their applications. The goal is to have a leisurely schedule, allowing plenty of time for informal discussions.

It is geared toward PhD students, and other researchers interested in the topic, but previous exposure to Lie groupoids will not be assumed.

The first day will contain either a small crash-course on Lie groupoids and Lie algebroids, or an open discussion with the participants, to introduce some definitions, examples, and ideas used in the mini-courses and the talks.

Mini-courses

• Lennart Obster (University of Coimbra): Fat Lie theory

• Wilmer Smilde (University of Illinois Urbana-Champaign): Algebroids and classification problems for geometric structures

• Francesco Cattafi and João Nuno Mestre: Crash-course on Lie groupoids and VB-groupoids

Talks

• Alejandro Cabrera (Universitat Politècnica de Catalunya): Incorporating symplectic groupoid multiplication into Poisson integrator methods

• Annika Kraasch-Tarnowsky (MPIM Bonn): Lie groupoids, connections and representations up to homotopy

• Francesco Cattafi (University of Würzburg): PB-groupoids and PB-algebroids

• Žan Grad (KU Leuven): On the obstructions to the existence of Cartan connections

Registration

Participation is free and open to all, but registration is required for logistical reasons.  We do not have, unfortunately, financial support for the participants. Please register by sending an email to jnmestre@mat.uc.pt saying that you want to participate.

Location

Department of Mathematics, University of Coimbra
Room 2.4

Abstracts

Lennart Obster

Title: Fat Lie theory

Abstract: The main goal of the course is to introduce the language of fat extensions for Lie groupoids and algebroids (although the focus will be on Lie groupoids). We start the course, however, with an introduction to abstract representations up to homotopy (ruths), which are the intrinsic objects behind usual (split) ruths. In doing so, we will revisit classical correspondences that are relevant (In particular, Dold-Kan and Serre-Swan), and formulate some interesting new directions to pursue (including some work in progress with Ioan Marcut and João Nuno Mestre). The main function of the introduction on ruths is so that we can treat the special class of 2-term ruths on equal footing with the many other equivalent ways to describe such objects. In fact, besides vector bundle groupoids (VB-groupoids) and general linear principal bundle groupoids (PB-groupoids), 2-term ruths are in a one-to-one correspondence with fat extensions.

After an introduction to fat extensions, where we focus on examples, we move to setting the above correspondences in terms of equivalences of categories (and 2-categories if time permits). Meanwhile, we make comments on related work in progress with João Nuno Mestre and Luca Vitagliano on multiplicative tensors.

While the equivalence between VB-groupoids and fat extensions will be used as a guide throughout, PB-groupoids will be discussed last, and feature a discussion on double groupoids as well. On the one hand, PB-groupoids come with a structural strict Lie 2-groupoid, which is a type of double groupoid (the main example is the general linear strict Lie 2-groupoid). On the other hand, PB-groupoids come with a gauge double groupoid, although such double groupoids are not always smooth. Studying cases when it is smooth reveals an interesting connection with objects that we call ``core extensions’’, a generalisation of the core diagrams of Brown, Jotz-Lean and Mackenzie. At the end of the course we will focus on the connection between core extensions, fat extensions and PB-groupoids.

Wilmer Smilde

Title:  Algebroids and classification problems for geometric structures

Abstract:

Alejandro Cabrera

Title: Incorporating symplectic groupoid multiplication into Poisson integrator methods

Abstract: This talk is based on recent joint work with D. Iglesias and J.C. Marrero. Following O. Cosserat, we first review a general approach to approximate hamiltonian flows in a Poisson manifold. Only the "strict bi-realization" data is used in these approaches, but not the underlying groupoid multiplication. We then explain how multiplication "m" can be incorporated into this construction as well as how generating functions for m yield recurrence "multiplicative" approximation formulas. We describe the resulting practical numerical algorithms and some illustrative examples.

Annika Kraasch-Tarnowsky

Title: Lie groupoids, connections and representations up to homotopy

Abstract: Lie groupoid representations, which are suitable Lie groupoid actions on a vector bundle over the space of objects, generalise the concept of Lie group representations. However, this definition is too narrow to allow for a generalisation of the adjoint and coadjoint representations, an issue which is solved by weakening the notion to a representation up to homotopy on a differential graded vector bundle. Representations up to homotopy have various applications, including deformation theory of Lie groupoids and infinitesimal models for differentiable stacks, as they are important tools to study the columns of the Bott-Shulman-Stasheff bicomplex. The coadjoint representation up to homotopy is constructed using a notion of connection on the source map of a Lie groupoid that is not necessarily compatible with the groupoid structure. However, there are various different notions of connections that have been used to study Lie groupoids, such as Cartan connections and, more recently, multiplicative Ehresmann connections, which both are retained by the groupoid multiplication. In this talk, the speaker will establish the definition of a representation up to homotopy, introduce the coadjoint representation up to homotopy, explore consequences of using a Cartan connection in its construction and connect the concept to the notion of $J^E G$-invariance, developed in the speaker's PhD thesis, which arises from a multiplicative Ehresmann connection $E$ on a regular Lie groupoid.

Francesco Cattafi

Title: PB-groupoids and PB-algebroids

Abstract: It is well known that the collection of linear frames of a vector bundle $E \to M$ of rank $k$ defines a principal $GL(k, R)$-bundle over $M$, called the frame bundle; conversely, any principal $GL(k, R)$-bundle induces an associated vector bundle via the attaching fibre construction. This correspondence allows one to transform objects and problems from one side to the other, and it can therefore be useful also when replacing manifolds with "higher structures". However, while the theory of "vector bundles over Lie groupoids" (VB-groupoids) and "vector bundles over Lie algebroids" (VB-algebroids) has been fairly developed in the past decades, little is known about the principal bundle counterpart of these objects.

In this talk, I will recall the classical VB-PB correspondence mentioned above, and then introduce a special class of frames of VB-groupoids which interact nicely with the groupoid structure. I will then use them to associate to any given VB-groupoid a diagram of Lie groupoids and principal bundles (in the appropriate sense); this will lead us to the general notion of a PB-groupoid, i.e. a "principal bundle over a Lie groupoid", whose structural object is a (strict) Lie 2-groupoid (i.e. a double Lie groupoid over the unit groupoid). In turn, this will generalise the standard correspondence between vector bundles and principal bundles to a correspondence between VB-groupoids and PB-groupoids.

I will then move to the infinitesimal picture and introduce the notion of PB-algebroid in two equivalent ways: by considering a special class of frames of VB-algebroids, and by differentiating a PB-groupoid. In this case, the structural object turns out to be not a (strict) Lie 2-algebroid, as one could first expect, but an LA-groupoid over the zero algebroid; I will explain how this is related to the differentiation/integration of double Lie groupoids and double Lie algebroids, and sketch the integration problem for PB-algebroids.

This is joint work (regarding PB-groupoids) and joint work in progress (regarding PB-algebroids) with Alfonso Garmendia. If time permits, I will conclude by mentioning future applications to geometric structures (current work in progress with Luca Accornero and Ivan Struchiner).

Žan Grad

Title: On the obstructions to the existence of Cartan connections

Abstract: Cartan connections provide a means of extending the adjoint and normal representation of a Lie groupoid G ⇉ M, from the isotropy and the normal bundle (of the orbit foliation) to the whole Lie algebroids A(G) and TM. As such, their existence is very restrictive, and this gives one motivation for the introduction of the notion of representations up to homotopy, central to our research. Detecting whether a Cartan connection exists is a nontrivial matter; in this talk, we will construct a Chern-Weil theory for short exact sequences of Lie algebroids, providing an obstruction in terms of characteristic classes which live in Lie algebroid cohomology, with coefficients in a representation up to homotopy.

This is joint work with Lennart Obster.

Organizers

• Raquel Caseiro

• João Nuno Mestre

Previous editions:

Lie groupoid and algebroid week in Coimbra III was held in June 2024.
Lie groupoid and algebroid week in Coimbra II was held in July 2023.

Lie groupoid week in Coimbra was held in September 2022.

Support:

CMUC - UID/00324/2025