Pseudo-group

of transformations of a differentiable manifold $ M $

A family of diffeomorphisms from open subsets of $ M $ into $ M $ that is closed under composition of mappings, transition to the inverse mapping, as well as under restriction and glueing of mappings. More precisely, a pseudo-group of transformations $ \Gamma $ of a manifold $ M $ consists of local transformations, i.e. pairs of the form $ p =( D _ {p} , \overline{p}\; ) $ where $ D _ {p} $ is an open subset of $ M $ and $ \overline{p}\; $ is a diffeomorphism $ D _ {p} \rightarrow M $, where it is moreover assumed that 1) $ p , q \in \Gamma $ implies $ p \circ q = ( \overline{q} ^ {- 1} ( D _ {p} \cap \overline{q} ( D _ {q} ) ) , \overline{p}\; \circ \overline{q}\; ) \in \Gamma $; 2) $ p \in \Gamma $ implies $ p ^ {- 1} = ( \overline{p} ( D _ {p} ) , \overline{p} ^ {- 1} ) \in \Gamma $; 3) $ ( M , \mathop{\rm id} ) \in \Gamma $; and 4) if $ \overline{p}\; $ is a diffeomorphism from an open subset $ D \subset M $ into $ M $ and $ D = \cup _ \alpha D _ \alpha $, where $ D _ \alpha $ are open sets in $ M $, then $ ( D , \overline{p} ) \in \Gamma \iff ( D _ \alpha , \overline{p}\; \mid _ {D _ \alpha } ) \in \Gamma $ for any $ \alpha $. With necessary changes in 1)–4) one can also define pseudo-groups of transformations of an arbitrary topological space (cf. [7]) or even of an arbitrary set. As a group of transformations, a pseudo-group of transformations determines an equivalence relation on $ M $; the equivalence classes are called its orbits. A pseudo-group $ \Gamma $ of transformations of a manifold $ M $ is called transitive if $ M $ is its only orbit, and is called primitive if $ M $ does not admit non-trivial $ \Gamma $-invariant foliations (otherwise the pseudo-group is called imprimitive).

A pseudo-group $ \Gamma $ of transformations of a differentiable manifold is called a Lie pseudo-group of transformations defined by a system $ S $ of partial differential equations if $ \Gamma $ consists of exactly those local transformations of $ M $ that satisfy the system $ S $. E.g., the pseudo-group of conformal transformations of the plane is a Lie pseudo-group of transformations, determined by the Cauchy–Riemann equations (cf. Cauchy-Riemann equations). The order of a Lie pseudo-group of transformations is the minimum order of its defining system of differential equations.

Examples of Lie pseudo-groups of transformations. a) The pseudo-group of all holomorphic local transformations of $ n $-dimensional complex space $ \mathbf C ^ {n} $.

b) The pseudo-group of all holomorphic local transformations of $ \mathbf C ^ {n} $ with constant Jacobian.

c) The pseudo-group of all holomorphic local transformations of $ \mathbf C ^ {n} $ with Jacobian 1.

d) The Hamilton pseudo-group of all holomorphic local transformations of $ \mathbf C ^ {n} $ ($ n $ even) preserving the differential 2-form

$$ \omega = d z ^ {1} \wedge d z ^ {2} + d z ^ {3} \wedge d z ^ {4} + \dots + d z ^ {n- 1} \wedge d z ^ {n} . $$

e) The pseudo-group of all holomorphic local transformations of $ \mathbf C ^ {n} $ preserving $ \omega $ up to constant factor.

f) The contact pseudo-group of all holomorphic local transformations of $ \mathbf C ^ {n} $ ($ n = 2 m + 1 $, $ m \geq 1 $) preserving the differential 1-form

$$ d z ^ {n} + \sum _ { i= 1} ^ { m } ( z ^ {i} d z ^ {m+ i} - z ^ {m+ i} d z ^ {i} ) $$

up to a factor (which can be a function).

g) The real analogues of the complex pseudo-groups of transformations of Examples a)–f).

The order of the Lie pseudo-groups of Examples a), c)–f) is 1, while in b) the order is 2.

Any Lie group $ G $ of transformations of a manifold $ M $ determines a pseudo-group $ \Gamma ( G) $ of transformations, consisting of the restrictions of the transformations from $ G $ onto open subsets of $ M $. A pseudo-group of transformations of the form $ \Gamma ( G) $ is called globalizable. E.g., a pseudo-group of local conformal transformations of the sphere $ S ^ {n} $ is globalizable for $ n > 2 $ and not globalizable for $ n = 2 $.

A Lie pseudo-group of transformations is said to be of finite type if there is a natural number $ d $ such that every local transformation $ p \in \Gamma $ is uniquely determined by its $ d $-jet at some point $ x \in D _ {p} $; the smallest such $ d $ is called the degree, or type, of $ \Gamma $; if such a $ d $ does not exist, then $ \Gamma $ is called a pseudo-group of transformations of infinite type. The pseudo-groups of Examples a)–f) are primitive Lie pseudo-groups of transformations of infinite type.

Let $ \Gamma $ be a transitive Lie pseudo-group of transformations of an $ n $-dimensional manifold $ M $ and let $ G ^ {r} ( \Gamma ) $ be the family of all $ r $-jets of the local transformations in $ \Gamma $ that preserve a point $ O \in M $, i.e. those $ p \in \Gamma $ for which $ O \in D _ {p} $ and $ \overline{p} ( O) = O $. The set $ G ^ {r} ( \Gamma ) $, endowed with the natural structure of a Lie group, is called the $ r $-th order isotropy group of $ \Gamma $ ($ G ^ {1} ( \Gamma ) $ is also called the linear isotropy group of $ \Gamma $). The Lie algebra $ \mathfrak g ^ {r} ( \Gamma ) $ of $ \Gamma ^ {r} ( \Gamma ) $ can be naturally imbedded in the Lie algebra of $ r $-jets of vector fields on $ M $ at $ O $. If $ \Gamma $ is a Lie pseudo-group of transformations of order one, then the kernel $ G ^ {( r)} ( \Gamma ) $ of the natural homomorphism $ G ^ {r+ 1} ( \Gamma ) \rightarrow G ^ {r} ( \Gamma ) $ depends, for any $ r \geq 1 $, only on the linear isotropy group $ G ^ {1} ( \Gamma ) $, and is called its $ r $-th extension. A Lie pseudo-group of transformations $ \Gamma $ of order one is of finite type $ d $ if and only if

$$ \mathop{\rm dim} G ^ {( d- 1)} ( \Gamma ) \neq 0 \ \ \textrm{ and } \ \ \mathop{\rm dim} G ^ {( d)} ( \Gamma ) = 0 . $$

If, moreover, $ G ^ {1} ( \Gamma ) $ is irreducible, then $ d \leq 2 $ (cf. ). A Lie pseudo-group of transformations $ \Gamma $ of order one is a pseudo-group of transformations of finite type only if, and in the complex case if and only if, the Lie algebra $ \mathfrak g ^ {1} $ does not contain endomorphisms of rank 1 (cf. [10]). Such linear Lie algebras are called elliptic.

One has calculated the Lie algebras of all extensions $ G ^ {( r)} ( \Gamma ) $, $ r \geq 1 $, where $ \Gamma $ is a Lie pseudo-group of transformations of order one, in terms of the linear isotropy algebra. More precisely, the Lie algebra $ \mathfrak g ^ {( r)} ( \Gamma ) $ of $ G ^ {( r)} ( \Gamma ) $ consists of the $ ( r+ 1) $-jets of vector fields on $ M $ at $ O $ having, in some local coordinate system $ ( x ^ {1} \dots x ^ {n} ) $, the form

$$ \sum v _ {i _ {0} \dots i _ {r} } ^ {i} x ^ {i _ {0} } \dots x ^ {i _ {r} } \frac \partial {\partial x ^ {i} } , $$

where $ v _ {i _ {0} \dots i _ {r} } ^ {i} $ is an arbitrary tensor that is symmetric with respect to the lower indices and that satisfies the condition: For any fixed $ i _ {1} \dots i _ {r} $ the matrix

$$ \| v _ {j , i _ {1} \dots i _ {r} } ^ {i} \| _ {i , j = 1 } ^ {n} $$

belongs to $ \mathfrak g ^ {1} ( \Gamma ) $, relative to some coordinate system $ ( x ^ {i} ) $.

Let $ M $ be an $ n $-dimensional differentiable manifold over the field $ K = \mathbf R $ or $ \mathbf C $. Every transitive Lie pseudo-group of transformations $ \Gamma $ of order $ k $ on a manifold $ M $ coincides with the pseudo-group of all local automorphism of some $ G ^ {k} ( \Gamma ) $-structure (cf. $ G $-structure) of order $ k $ on $ M $ (Cartan's first fundamental theorem). The first classification of all primitive Lie pseudo-groups of infinite type was obtained by E. Cartan . According to his theorem, every primitive Lie pseudo-group of transformations of infinite type, consisting of holomorphic local transformations, is locally isomorphic to one of the pseudo-groups of Examples a)–f). This theorem has been repeatedly proved; its modern proofs lead to the study of certain filtered Lie algebras (cf. [9]). The classification of these filtered Lie algebras can be given on the basis of the classification of simple graded Lie algebras (cf. [3]). The classification of primitive pseudo-groups of transformations has also been obtained in the real case, and the condition of analyticity of the action of the pseudo-group of transformations has been replaced by the weaker condition of infinite differentiability (cf. [8], [9]). One has constructed certain abstract models of transitive Lie pseudo-groups, which came to play the same role in the theory of pseudo-groups of transformations of infinite type as do abstract Lie groups in the finite-dimensional case (cf. , [9]).

References

[1]	S. Sternberg, "Lectures on differential geometry" , Prentice-Hall (1964) MR0193578 Zbl 0129.13102
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[2c]	E. Cartan, "Les groupes de transformations continus, infinis, simples" , Oeuvres complètes , 2 , Gauthier-Villars (1953) pp. 857–925 MR1509105 Zbl 40.0193.02 Zbl 38.0194.01
[2d]	E. Cartan, "Les groupes de transformations continus, infinis, simples" , Oeuvres complètes , 2 , Gauthier-Villars (1953) pp. 1335–1384 MR1509105 Zbl 40.0193.02 Zbl 38.0194.01
[3]	V. Guillemin, "Infinite dimensional primitive Lie algebras" J. Diff. Geom. , 4 : 3 (1970) pp. 257–282 MR0268233 Zbl 0223.17007
[4]	S. Kobayashi, "Transformation groups in differential geometry" , Springer (1972) MR0355886 Zbl 0246.53031
[5a]	S. Kobayashi, T. Nagano, "On filtered Lie algebras and geometric structures I" J. Math. Mech. , 13 : 5 (1964) pp. 875–907 MR0168704 Zbl 0142.19504
[5b]	S. Kobayashi, T. Nagano, "On filtered Lie algebras and geometric structures III" J. Math. Mech. , 14 : 5 (1965) pp. 679–706 MR0188364
[6a]	M. Kuranishi, "On the local theory of continuous infinite pseudo groups I" Nagoya Math. J. , 15 (1959) pp. 225–260 MR0116071 Zbl 0212.56501
[6b]	M. Kuranishi, "On the local theory of continuous infinite pseudo groups II" Nagoya Math. J. , 19 (1961) pp. 55–91 MR0142694 Zbl 0212.56501
[7]	P. Libermann, "Pseudogroupes infinitésimaux attachées aux pseudogroupes de Lie" Bull. Soc. Math. France , 87 : 4 (1959) pp. 409–425 MR123279
[8]	S. Shnider, "The classification of real primitive infinite Lie algebras" J. Diff. Geom. , 4 : 1 (1970) pp. 81–89 MR0285574 Zbl 0244.17014
[9]	I.M. Singer, S. Sternberg, "The infinite groups of Lie and Cartan. I. The transitive groups" J. d'Anal. Math. , 15 (1965) pp. 1–114 MR0217822 Zbl 0277.58008
[10]	R.L. Wilson, "Irreducible Lie algebras of infinite type" Proc. Amer. Math. Soc. , 29 : 2 (1971) pp. 243–249 MR0277582 Zbl 0216.07401

References

[a1]	C. Albert, P. Molino, "Pseudogroupes de Lie transitifs" , I–II , Hermann (1984–1987) MR0904048 MR0770061 Zbl 0682.53003 Zbl 0563.53027
[a2]	J.F. Pommaret, "Systems of partial differential equations and Lie pseudogroups" , Gordon & Breach (1978) MR0517402 Zbl 0418.35028 Zbl 0401.58006