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bisection of a Lie groupoid

Context

\infty-Lie theory

∞-Lie theory (higher geometry)

Background

Smooth structure

Higher groupoids

Lie theory

∞-Lie groupoids

∞-Lie algebroids

Formal Lie groupoids

Cohomology

Homotopy

Examples

\infty-Lie groupoids

\infty-Lie groups

\infty-Lie algebroids

\infty-Lie algebras

Bisections of Lie groupoids

Definition

In components

Definition

Let (X 1(d 0,d 1)X 0×X 0)(X_1 \stackrel{(d_0, d_1)}{\to} X_0 \times X_0) be a Lie groupoid.

A bisection of is a smooth function σ:X 0X 1\sigma : X_0 \to X_1 such that

  1. σ\sigma is a section of d 1d_1;

  2. d 0σ:X 0X 0d_0 \circ \sigma : X_0 \to X_0 is a diffeomorphism.

Bisections naturally form a group under pointwise composition in XX, 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 H=\mathbf{H} = Smooth∞Grpd. Let XHX \in \mathbf{H} be equipped with an atlas, hence with an effective epimorphism UXU \to X out of a 0-truncated object.

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

Definition

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

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

For XX a 1-groupoid as above and U=X 0U = X_0, a bisection is precisely a smooth natural transformation of the form

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

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

x(xη(x)ϕ(x)). 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 UXU \to X a Lie groupoid with atlas as above, write 𝔤=Lie(BiSect(X,U))\mathfrak{g} = Lie(\mathbf{BiSect}(X,U)) for the Lie algebra of the group of bisections. Then (C (X),𝔤)(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

Last revised on May 30, 2018 at 04:45:46. See the history of this page for a list of all contributions to it.