In mathematics, a pseudogroup is a set of diffeomorphisms between open sets of a space, satisfying group-like and sheaf-like properties. It is a generalisation[dubious – discuss] of the concept of a group, originating however from the geometric approach of Sophus Lie[1] 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.[2][3]
Definition
editA pseudogroup imposes several conditions on sets of homeomorphisms (respectively, diffeomorphisms) defined on open sets U of a given Euclidean space or more generally of a fixed topological space (respectively, smooth manifold). Since two homeomorphisms h : U → V and g : V → W compose to a homeomorphism from U to W, 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 S is a collection
- The domains of the elements g in
Γ cover S ("cover"). - The restriction of an element g in
Γ to any open set contained in its domain is also inΓ ("restriction"). - The composition g ∘ h of two elements of
Γ , when defined, is inΓ ("composition"). - The inverse of an element of g is in
Γ ("inverse"). - The property of lying in
Γ is local, i.e. if g : U → V is a homeomorphism between open sets of S and U is covered by open sets Ui with g restricted to Ui lying inΓ for each i, then g also lies inΓ ("local").
As a consequence the identity homeomorphism of any open subset of S lies in
Similarly, a pseudogroup on a smooth manifold X is defined as a collection
Two points in X are said to be in the same orbit if an element of
Examples
editA widespread class of examples is given by pseudogroups preserving a given geometric structure. For instance, if (X, g) is a Riemannian manifold, one has the pseudogroup of its local isometries; if (X,
Pseudogroups of symmetries and geometric structures
editManifolds with additional structures can often be defined using the pseudogroups of symmetries of a fixed local model. More precisely, given a pseudogroup
In particular, when
- flat Riemannian structures, for
Γ pseudogroups of isometries of Rn with the canonical Euclidean metric; - symplectic structures, for
Γ the pseudogroup of symplectomorphisms of R2n with the canonical symplectic form; - analytic structures, for
Γ the pseudogroup of (real-)analytic diffeomorphisms of Rn; - Riemann surfaces, for
Γ the pseudogroup of invertible holomorphic functions of a complex variable.
More generally, any integrable G-structure and any (G, X)-manifold are special cases of
Pseudogroups and Lie theory
editIn 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[7] and D. C. Spencer.[8] 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;[9][10][11][12][13] 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
References
edit- ^ Sophus, Lie (1888–1893). Theorie der Transformationsgruppen. B.G. Teubner. OCLC 6056947.
- ^ Cartan, Élie (1904). "Sur la structure des groupes infinis de transformations" (PDF). Annales Scientifiques de l'École Normale Supérieure. 21: 153–206. doi:10.24033/asens.538.
- ^ Cartan, Élie (1909). "Les groupes de transformations continus, infinis, simples" (PDF). Annales Scientifiques de l'École Normale Supérieure. 26: 93–161. doi:10.24033/asens.603.
- ^ Kobayashi, Shoshichi; Nomizu, Katsumi (1963). Foundations of Differential Geometry, Volume I. Wiley Classics Library. New York: John Wiley & Sons Inc. pp. 1–2. ISBN 0470496487.
- ^ Thurston, William P. (1997). Silvio Levy (ed.). Three-dimensional geometry and topology. Princeton Mathematical Series. Vol. 35. Princeton University Press. doi:10.1515/9781400865321. ISBN 0-691-08304-5. MR 1435975.
- ^ Loomis, Lynn; Sternberg, Shlomo (2014). "Differentiable manifolds". Advanced Calculus (Revised ed.). World Scientific. pp. 364–372. ISBN 978-981-4583-93-0. MR 3222280.
- ^ Kodaira, K. (1960). "On Deformations of Some Complex Pseudo-Group Structures". Annals of Mathematics. 71 (2): 224–302. doi:10.2307/1970083. ISSN 0003-486X. JSTOR 1970083.
- ^ Guillemin, Victor; Sternberg, Shlomo (1966). "Deformation theory of pseudogroup structures". Memoirs of the American Mathematical Society (64): 0. doi:10.1090/memo/0064. ISSN 0065-9266.
- ^ Kumpera, Antonio; Spencer, Donald Clayton (1973-01-01). Lie Equations, Vol. I. Princeton University Press. doi:10.1515/9781400881734. ISBN 978-1-4008-8173-4.
- ^ Singer, I. M.; Sternberg, Shlomo (1965). "The infinite groups of Lie and Cartan Part I, (The transitive groups)". Journal d'Analyse Mathématique. 15 (1): 1–114. doi:10.1007/bf02787690. ISSN 0021-7670. S2CID 123124081.
- ^ Claude., Albert (1984–1987). Pseudogroupes de Lie transitifs. Hermann. OCLC 715985799.
- ^ Kuranishi, Masatake (1959). "On the Local Theory of Continuous Infinite Pseudo Groups I". Nagoya Mathematical Journal. 15: 225–260. doi:10.1017/s0027763000006747. ISSN 0027-7630.
- ^ Olver, Peter J.; Pohjanpelto, Juha (2005). "Maurer–Cartan forms and the structure of Lie pseudo-groups". Selecta Mathematica. 11 (1): 99–126. doi:10.1007/s00029-005-0008-7. ISSN 1022-1824. S2CID 14712181.
- St. Golab (1939). "Über den Begriff der "Pseudogruppe von Transformationen"". Mathematische Annalen. 116: 768–780. doi:10.1007/BF01597390. S2CID 124962440.
External links
edit- Alekseevskii, D.V. (2001) [1994], "Pseudo-groups", Encyclopedia of Mathematics, EMS Press