Pseudogroup

Template:Short description In mathematics, a pseudogroup is a set of homeomorphisms between open sets of a space, satisfying group-like and sheaf-like properties. It is a generalisation{{ safesubst:#invoke:Unsubst||date=__DATE__ |$B= Template:Fix }} of the concept of a transformation group, originating however from the geometric approach of Sophus Lie<ref>Template:Cite book</ref> to investigate symmetries of differential equations, rather than out of abstract algebra (such as quasigroup, for example). The modern theory of pseudogroups was developed by Élie Cartan in the early 1900s.<ref>Template:Cite journal</ref><ref>Template:Cite journal</ref>

Definition

A pseudogroup imposes several conditions on sets of homeomorphisms (respectively, diffeomorphisms) defined on open sets Template:Math of a given Euclidean space or more generally of a fixed topological space (respectively, smooth manifold). Since two homeomorphisms Template:Math and Template:Math compose to a homeomorphism from Template:Math to Template:Math, one asks that the pseudogroup is closed under composition and inversion. However, unlike those for a group, the axioms defining a pseudogroup are not purely algebraic; the further requirements are related to the possibility of restricting and of patching homeomorphisms (similar to the gluing axiom for sections of a sheaf).

More precisely, a pseudogroup on a topological space Template:Math is a collection Template:Math of homeomorphisms between open subsets of Template:Math satisfying the following properties:<ref name="KN">Template:Cite book</ref><ref name="Thurston">Template:Cite book</ref>

1. The domains of the elements Template:Math in Template:Math cover Template:Math ("cover").
2. The restriction of an element Template:Math in Template:Math to any open set contained in its domain is also in Template:Math ("restriction").
3. The composition Template:Math of two elements of Template:Math, when defined, is in Template:Math ("composition").
4. The inverse of an element of Template:Math is in Template:Math ("inverse").
5. The property of lying in Template:Math is local, i.e. if Template:Math is a homeomorphism between open sets of Template:Math and Template:Math is covered by open sets Template:Math with Template:Math restricted to Template:Math lying in Template:Math for each Template:Math, then Template:Math also lies in Template:Math ("local").

As a consequence the identity homeomorphism of any open subset of Template:Math lies in Template:Math.

Similarly, a pseudogroup on a smooth manifold Template:Math is defined as a collection Template:Math of diffeomorphisms between open subsets of Template:Math satisfying analogous properties (where we replace homeomorphisms with diffeomorphisms).<ref>Template:Cite book</ref>

Two points in Template:Math are said to be in the same orbit if an element of Template:Math sends one to the other. Orbits of a pseudogroup clearly form a partition of Template:Math; a pseudogroup is called transitive if it has only one orbit.

Examples

A widespread class of examples is given by pseudogroups preserving a given geometric structure. For instance, if Template:Math is a Riemannian manifold, one has the pseudogroup of its local isometries; if Template:Math is a symplectic manifold, one has the pseudogroup of its local symplectomorphisms; etc. These pseudogroups should be thought as the set of the local symmetries of these structures.

Pseudogroups of symmetries and geometric structures

Manifolds with additional structures can often be defined using the pseudogroups of symmetries of a fixed local model. More precisely, given a pseudogroup Template:Math, a [[Atlas (topology)|Template:Math-atlas]] on a topological space Template:Math consists of a standard atlas on Template:Math such that the changes of coordinates (i.e. the transition maps) belong to Template:Math. An equivalent class of Γ-atlases is also called a Template:Math-structure on Template:Math.

In particular, when Template:Math is the pseudogroup of all locally defined diffeomorphisms of Template:Math, one recovers the standard notion of a smooth atlas and a smooth structure. More generally, one can define the following objects as Template:Math-structures on a topological space Template:Math:

• flat Riemannian structures, for Template:Math pseudogroups of isometries of Template:Math with the canonical Euclidean metric;
• symplectic structures, for Template:Math the pseudogroup of symplectomorphisms of Template:Math with the canonical symplectic form;
• analytic structures, for Template:Math the pseudogroup of (real-)analytic diffeomorphisms of Template:Math;
• Riemann surfaces, for Template:Math the pseudogroup of invertible holomorphic functions of a complex variable.

More generally, any integrable [[G-structure on a manifold|Template:Math-structure]] and any [[(G,X)-manifold|Template:Math-manifold]] are special cases of Template:Math-structures, for suitable pseudogroups Template:Math.

Pseudogroups and Lie theory

In general, pseudogroups were studied as a possible theory of infinite-dimensional Lie groups. The concept of a local Lie group, namely a pseudogroup of functions defined in neighbourhoods of the origin of a Euclidean space Template:Math, is actually closer to Lie's original concept of Lie group, in the case where the transformations involved depend on a finite number of parameters, than the contemporary definition via manifolds. One of Cartan's achievements was to clarify the points involved, including the point that a local Lie group always gives rise to a global group, in the current sense (an analogue of Lie's third theorem, on Lie algebras determining a group). The formal group is yet another approach to the specification of Lie groups, infinitesimally. It is known, however, that local topological groups do not necessarily have global counterparts.

Examples of infinite-dimensional pseudogroups abound, beginning with the pseudogroup of all diffeomorphisms of Template:Math. The interest is mainly in sub-pseudogroups of the diffeomorphisms, and therefore with objects that have a Lie algebra analogue of vector fields. Methods proposed by Lie and by Cartan for studying these objects have become more practical given the progress of computer algebra.

In the 1950s, Cartan's theory was reformulated by Shiing-Shen Chern, and a general deformation theory for pseudogroups was developed by Kunihiko Kodaira<ref>Template:Cite journal</ref> and D. C. Spencer.<ref>Template:Cite journal</ref> In the 1960s homological algebra was applied to the basic PDE questions involved, of over-determination; this though revealed that the algebra of the theory is potentially very heavy. In the same decade the interest for theoretical physics of infinite-dimensional Lie theory appeared for the first time, in the shape of current algebra.

Intuitively, a Lie pseudogroup should be a pseudogroup which "originates" from a system of PDEs. There are many similar but inequivalent notions in the literature;<ref>Template:Cite book</ref><ref>Template:Cite journal</ref><ref>Template:Cite book</ref><ref>Template:Cite journal</ref><ref>Template:Cite journal</ref> the "right" one depends on which application one has in mind. However, all these various approaches involve the (finite- or infinite-dimensional) jet bundles of Template:Math, which are asked to be a Lie groupoid. In particular, a Lie pseudogroup is called of finite order Template:Math if it can be "reconstructed" from the space of its Template:Math-jets.

References

• Template:Cite journal

External links

• Template:Springer