- Pseudogroup
In
mathematics , a pseudogroup is an extension of the group concept, but one that grew out of the geometric approach ofSophus Lie , rather than out ofabstract algebra (such asquasigroup , for example). A theory of pseudogroups was developed byÉlie Cartan in the early 1900s. [cite journal|first = Élie|last = Cartan|title = [http://archive.numdam.org/article/ASENS_1904_3_21__153_0.pdf Sur la structure des groupes infinis de transformations] |journal = Annales scientifiques de l'É.N.S.|year = 1904|volume = 21|pages=153–206] [cite journal|first = Élie|last = Cartan|title = [http://archive.numdam.org/article/ASENS_1909_3_26__93_0.pdf Les groupes de transformations continus, infinis, simples] |journal = Annales scientifiques de l'École Normale Supérieure Sér. 3|year = 1909|volume = 26|pages=93–161]It is not an axiomatic algebraic idea; rather it defines a set of closure conditions on sets of
homeomorphism s defined onopen set s "U" of a givenEuclidean space "E" or more generally of a fixedtopological space "S". Thegroupoid 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 function s, or agluing axiom ).Specifically, a pseudogroup on a topological space "S" is a collection "Γ" of homeomorphisms between open subsets of "S" satisfying the following properties. [Kobayashi, Shoshichi & Nomizu, Katsumi. Foundations of Differential Geometry, Volume I. Wiley Classics Library. John Wiley & Sons Inc., New York, 1996. Reprint of the 1969 original, A Wiley-Interscience Publication. ISBN 0-471-15733-3.]
* 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 "{ Ui }", "f" is a homeomorphism from "U" to an open subset of "S", and the restriction of "f" to "Ui" is in "Γ" for all "i", then "f" is in "Γ".
* If "f":"U" → "V" and "f ′":"U ′" → "V ′" are in "Γ", and theintersection "V ∩ U ′" is not empty, then the following restricted composition is in "Γ"::f' circ f colon f^{-1}(V cap U') o f'(V cap U').An example in space of two dimensions is the pseudogroup of invertible
holomorphic function s of acomplex variable (invertible in the sense of having aninverse function ). The properties of this pseudogroup are what makes it possible to defineRiemann surface s by local data patched together.In general, pseudogroups were studied as a possible theory of
infinite-dimensional Lie group s. The concept of a local Lie group, namely a pseudogroup of functions defined in neighbourhoods of the origin of "E", is actually closer to Lie's original concept ofLie group , in the case where the transformations involved depend on a finite number ofparameter s, than the contemporary definition viamanifold s. 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 ofLie's third theorem , onLie algebra s determining a group). Theformal group is yet another approach to the specification of Lie groups, infinitesimally. It is known, however, that "localtopological group s" do not necessarily have global counterparts.Examples of infinite-dimensional pseudogroups abound, beginning with the pseudogroup of all
diffeomorphism s of "E". The interest is mainly in sub-pseudogroups of the diffeomorphisms, and therefore with objects that have a Lie algebra analogue ofvector field s. Methods proposed by Lie and by Cartan for studying these objects have become more practical given the progress ofcomputer algebra .In the 1950s Cartan's theory was reformulated by
Shiing-Shen Chern , and a generaldeformation theory for pseudogroups was developed byKunihiko Kodaira andD. C. Spencer . In the 1960shomological 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 fortheoretical physics of infinite-dimensional Lie theory appeared for the first time, in the shape ofcurrent algebra .References
External links
*springer|id=p/p075710|title=Pseudo-groups|author=Alekseevskii, D.V.
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