nLab bisection of a Lie groupoid

Definition

In components

Definition

Let $X=(X_1 \stackrel{(d_0, d_1)}{\to} X_0 \times X_0)$ be a Lie groupoid.

A bisection of $X$ is a smooth function $\sigma : X_0 \to X_1$ such that

1. $\sigma$ is a section of $d_1$;

2. $d_0 \circ \sigma : X_0 \to X_0$ is a diffeomorphism.

Bisections naturally form a group under pointwise composition in $X$, the group of bisections of the Lie groupoid.

One can prove that the bisection group is a infinite-dimensional Lie group in the sense of Milnor (see Neeb’s survey) (under some mild assumptions on the underlying Lie groupoid). The infinite-dimensional Lie group of bisections is closely connected to the underlying Lie groupoid (see references below), e.g.

1. From the knowledge of the smooth structure of the bisection group and the manifold of units, one can even reconstruct the underlying Lie groupoid (again under some assumptions).

2. The construction is functorial in a suitable sense and extending this one can even relate (smooth) representations of Lie groupoids to smooth representations of its bisection group

Abstractly

Let $\mathbf{H} =$ Smooth∞Grpd. Let $X \in \mathbf{H}$ be equipped with an atlas, hence with an effective epimorphism $U \to X$ out of a 0-truncated object.

We may regard this atlas as an object in the slice (∞,1)-topos $\mathbf{X} \in \mathbf{H}_{/X}$

Definition

The smooth ∞-group of bisections of $\mathbf{X}$ is its automorphism ∞-group

$$\mathbf{BiSect}(X,U) \coloneqq \mathbf{Aut}_{/X}(\mathbf{X}, \mathbf{X}) \,.$$

Remark

For $X$ a 1-groupoid as above and $U = X_0$, a bisection is precisely a smooth natural transformation of the form

$$\array{ U &&\stackrel{\simeq}{\to}&& U \\ & \searrow &\swArrow_{\mathrlap{\eta}}& \swarrow \\ && X } \,.$$

Here the top morphism is a diffeomorphism $\phi : X \to X$ and since the diagonal morphisms are identities onto the object manifold the component map of $\eta$ is

$$x \mapsto (x \stackrel{\eta(x)}{\to} \phi(x)) \,.$$

This is precisely the bisection in the traditional sense of def. .

Properties

Relation to Lie-Rinehart algebras

For $U \to X$ a Lie groupoid with atlas as above, write $\mathfrak{g} = Lie(\mathbf{BiSect}(X,U))$ for the Lie algebra of the group of bisections. Then $(C^\infty(X), \mathfrak{g})$ is the Lie-Rinehart algebra corresponding to the Lie algebroid of the Lie groupoid.

Relation to Atiyah groupoids

for the moment see at Atiyah groupoid and higher Atiyah groupoid.

References

• Ieke Moerdijk, Janez Mrčun, p. 114 of Introduction to foliations and Lie groupoids, Cambridge Studies in Advanced Mathematics 91, 2003. x+173 pp. ISBN: 0-521-83197-0

• Alexander Schmeding, Christoph Wockel, The Lie group of bisections of a Lie groupoid (arXiv:1409.1428)

• Alexander Schmeding, Christoph Wockel, (Re)constructing Lie groupoids from their bisections and applications to prequantisation (arXiv:1506.05415)

• Alexander Schmeding, Christoph Wockel, Functorial aspects of the reconstruction of Lie groupoids from their bisections (arXiv:1506.05587)

• Habib Amiri, Alexander Schmeding, Linking Lie groupoid representations and representations of infinite-dimensional Lie groups (arXiv:1805.03935)