Pseudogroup

In mathematics, a pseudogroup is an extension of the group concept, but one that grew out of the geometric approach of Sophus Lie, rather than out of abstract algebra (such as quasigroup, for example). A theory of pseudogroups was developed by Élie Cartan in the early 1900s.[1][2]

It is not an axiomatic algebraic idea; rather it defines a set of closure conditions on sets of homeomorphisms defined on open sets U of a given Euclidean space E or more generally of a fixed topological space S. The groupoid condition on those is fulfilled, in that homeomorphisms h : U → V and g : V → W compose to a homeomorphism from U to W. The further requirement on a pseudogroup is related to the possibility of patching (in the sense of descent, transition functions, or a gluing axiom).

Specifically, a pseudogroup on a topological space S is a collection Γ of homeomorphisms between open subsets of S satisfying the following properties.[3]

• For every open set U in S, the identity map on U is in Γ.
• If f is in Γ, then so is f⁻¹.
• If f is in Γ, then the restriction of f to an arbitrary open subset of its domain is in Γ.
• If U is open in S, U is the union of the open sets {U_i}, f is a homeomorphism from U to an open subset of S, and the restriction of f to U_i is in Γ for all i, then f is in Γ.
• If f : U → V and f′ : U′ → V′ are in Γ, and the intersection V ∩ U′ is non-empty, then the following restricted composition is in Γ:

${\displaystyle f'\circ f\colon f^{-1}(V\cap U')\to f'(V\cap U').}$

An example in space of two dimensions is the pseudogroup of invertible holomorphic functions of a complex variable. The properties of this pseudogroup are what makes it possible to define Riemann surfaces by local data patched together.

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 E, 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 E. 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 and D. C. Spencer. 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.