∞-Lie theory (higher geometry)
Background
Smooth structure
Higher groupoids
Lie theory
∞-Lie groupoids
∞-Lie algebroids
Formal Lie groupoids
Cohomology
Homotopy
Related topics
Examples
$\infty$-Lie groupoids
$\infty$-Lie groups
$\infty$-Lie algebroids
$\infty$-Lie algebras
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
$\sigma$ is a section of $d_1$;
$d_0 \circ \sigma : X_0 \to X_0$ 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.
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).
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
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 $\mathbf{X} \in \mathbf{H}_{/X}$
The smooth ∞-group of bisections of $\mathbf{X}$ is its automorphism ∞-group
For $X$ a 1-groupoid as above and $U = X_0$, a bisection is precisely a smooth natural transformation of the form
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
This is precisely the bisection in the traditional sense of def. .
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.
for the moment see at Atiyah groupoid and higher Atiyah groupoid.
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)
Last revised on January 26, 2024 at 16:16:57. See the history of this page for a list of all contributions to it.