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| In [[mathematics]], a '''pseudogroup''' is an extension of the [[group (mathematics)|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.<ref>{{cite journal|first = Élie|last = Cartan|title = Sur la structure des groupes infinis de transformations|journal = Annales Scientifiques de l'École Normale Supérieure|year = 1904|volume = 21|pages=153–206|url=http://archive.numdam.org/article/ASENS_1904_3_21__153_0.pdf}}</ref><ref>{{cite journal|first = Élie|last = Cartan|title = Les groupes de transformations continus, infinis, simples|journal = Annales Scientifiques de l'École Normale Supérieure|year = 1909|volume = 26|pages=93–161|url=http://archive.numdam.org/article/ASENS_1909_3_26__93_0.pdf}}</ref>
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| It is not an axiomatic algebraic idea; rather it defines a set of closure conditions on sets of [[homeomorphism]]s defined on [[open set]]s ''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
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| :''h'':''U'' → ''V''
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| and
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| :''g'':''V'' → ''W''
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| 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 (category theory)|descent]], [[transition function]]s, or a [[gluing axiom]]).
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| Specifically, a '''pseudogroup''' on a topological space ''S'' is a collection ''Γ'' of homeomorphisms between open subsets of ''S'' satisfying the following properties.<ref>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.</ref>
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| * For every open set ''U'' in ''S'', the identity map on ''U'' is in ''Γ''.
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| * If ''f'' is in ''Γ'', then so is ''f <sup>−1</sup>''.
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| * If ''f'' is in ''Γ'', then the restriction of ''f'' to an arbitrary open subset of its domain is in ''Γ''.
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| * If ''U'' is open in ''S'', ''U'' is the union of the open sets ''{ U<sub>i</sub> }'', ''f'' is a homeomorphism from ''U'' to an open subset of ''S'', and the restriction of ''f'' to ''U<sub>i</sub>'' is in ''Γ'' for all ''i'', then ''f'' is in ''Γ''.
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| * If ''f'':''U'' → ''V'' and ''f ′'':''U ′'' → ''V ′'' are in ''Γ'', and the [[Intersection (set theory)|intersection]] ''V ∩ U ′'' is not empty, then the following restricted composition is in ''Γ'':
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| :<math>f' \circ f \colon f^{-1}(V \cap U') \to f'(V \cap U')</math>.
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| An example in space of two dimensions is the pseudogroup of invertible [[holomorphic function]]s of a [[complex variable]] (invertible in the sense of having an [[inverse function]]). The properties of this pseudogroup are what makes it possible to define [[Riemann surface]]s by local data patched together.
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| 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 of [[Lie group]], in the case where the transformations involved depend on a finite number of [[parameter]]s, than the contemporary definition via [[manifold]]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 of [[Lie's third theorem]], on [[Lie algebra]]s determining a group). The [[formal group]] is yet another approach to the specification of Lie groups, infinitesimally. It is known, however, that ''local [[topological group]]s'' do not necessarily have global counterparts.
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| 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 of [[vector field]]s. Methods proposed by Lie and by Cartan for studying these objects have become more practical given the progress of [[computer algebra]].
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| 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 [[partial differential equation|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]].
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| == References ==
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| <references/>
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| *{{cite journal | author=St. Golab | title=Über den Begriff der "Pseudogruppe von Transformationen" | journal=Mathematische Annalen | year=1939 | volume=116 | pages=768–780 | doi=10.1007/BF01597390}}
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| == External links == | |
| *{{springer|id=p/p075710|title=Pseudo-groups|author=Alekseevskii, D.V.}}
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| [[Category:Lie groups]]
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| [[Category:Algebraic structures]]
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| [[Category:Differential geometry]]
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| [[Category:Differential topology]]
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