arXiv subjects

Habib Amiri

Publications and source records attributed to Habib Amiri.

4 recordsLinked to original sources

arXiv2024

A differentiable monoid of smooth maps on Lie groupoids

In this article we investigate a monoid of smooth mappings on the space of arrows of a Lie groupoid and its group of units. The group of units turns out to be an infinite-dimensional Lie group which is regular in the sense of Milnor. Furthermore, this group is closely connected to the group of bisections of the Lie groupoid. Under suitable conditions, i.e. the source map of the Lie groupoid is proper, one also obtains a differentiable structure on the monoid and can identify the bisection group as a Lie subgroup of its group of units. Finally, relations between groupoids associated to the underlying Lie groupoid and subgroups of the monoid are obtained. The key tool driving the investigation is a generalisation of a result by A. Stacey which we establish in the present article. This result, called the Stacey-Roberts Lemma, asserts that pushforwards of submersions yield submersions between the infinite-dimensional manifolds of mappings.

math.GR↗

arXiv2019

Lie groupoids of mappings taking values in a Lie groupoid

Endowing differentiable functions from a compact manifold to a Lie group with the pointwise group operations one obtains the so-called current groups and, as a special case, loop groups. These are prime examples of infinite-dimensional Lie groups modelled on locally convex spaces. In the present paper, we generalise this construction and show that differentiable mappings on a compact manifold (possibly with boundary) with values in a Lie groupoid form infinite-dimensional Lie groupoids which we call current Lie groupoids. We then study basic differential geometry and Lie theory for these Lie groupoids of mappings. In particular, we show that certain Lie groupoid properties, like being a proper étale Lie groupoid, are inherited by the current groupoid. Furthermore, we identify the Lie algebroid of a current Lie groupoid as a current Lie algebroid (analogous to the current Lie algebra associated to a current Lie group). To establish these results, we study superposition operators given by postcomposition with a fixed function, between manifolds of $C^\ell$-functions. Under natural hypotheses, these operators turn out to be a submersion (an immersion, an embedding, proper, resp., a local diffeomorphism) if so is the underlying map. These results are new in their generality and of independent interest.

math.DG↗

arXiv2019

Linking Lie groupoid representations and representations of infinite-dimensional Lie groups

The present paper links the representation theory of Lie groupoids and infinite-dimensional Lie groups. We show that smooth representations of Lie groupoids give rise to smooth representations of associated Lie groups. The groups envisaged here are the bisection group and a group of groupoid self maps. Then representations of the Lie groupoids give rise to representations of the infinite-dimensional Lie groups on spaces of (compactly supported) bundle sections. Endowing the spaces of bundle sections with a fine Whitney type topology, the fine very strong topology, we even obtain continuous and smooth representations. It is known that in the topological category, this correspondence can be reversed for certain topological groupoids. We extend this result to the smooth category under weaker assumptions on the groupoids.

math.GR↗

arXiv2013

A left topological monoid associated to a topological groupoid

This paper presents a fanctor $S$ from the category of groupoids to the category of semigroups. Indeed, a monoid $S_G$ with a right zero element is related to a topological groupoid $G$. The monoid $S_G$ is a subset of $C(G,G)$, the set of all continuous functions from $G$ to $G$, and with the compact- open topology inherited from C(G,G) is a left topological monoid. The group of units of $S_G$, which is denoted by $H(1)$, is isomorphic to a subgroup of the group of all bijection map from $G$ to $G$ under composition of functions. Moreover, it is proved that $H(1)$ is embedded in the group of all invertible linear operators on $C(G)$, the set of all complex continuous function on $G$.

math.CT↗