View source for Pseudo-group

''of transformations of a differentiable manifold <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075700/p0757001.png" />''

A family of diffeomorphisms from open subsets of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075700/p0757002.png" /> into <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075700/p0757003.png" /> 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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075700/p0757004.png" /> of a manifold <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075700/p0757005.png" /> consists of local transformations, i.e. pairs of the form <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075700/p0757006.png" /> where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075700/p0757007.png" /> is an open subset of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075700/p0757008.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075700/p0757009.png" /> is a [[Diffeomorphism|diffeomorphism]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075700/p07570010.png" />, where it is moreover assumed that 1) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075700/p07570011.png" /> implies <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075700/p07570012.png" />; 2) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075700/p07570013.png" /> implies <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075700/p07570014.png" />; 3) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075700/p07570015.png" />; and 4) if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075700/p07570016.png" /> is a diffeomorphism from an open subset <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075700/p07570017.png" /> into <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075700/p07570018.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075700/p07570019.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075700/p07570020.png" /> are open sets in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075700/p07570021.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075700/p07570022.png" /> for any <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075700/p07570023.png" />. With necessary changes in 1)–4) one can also define pseudo-groups of transformations of an arbitrary topological space (cf. [[#References|[7]]]) or even of an arbitrary set. As a group of transformations, a pseudo-group of transformations determines an equivalence relation on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075700/p07570024.png" />; the equivalence classes are called its orbits. A pseudo-group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075700/p07570025.png" /> of transformations of a manifold <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075700/p07570026.png" /> is called transitive if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075700/p07570027.png" /> is its only orbit, and is called primitive if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075700/p07570028.png" /> does not admit non-trivial <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075700/p07570029.png" />-invariant foliations (otherwise the pseudo-group is called imprimitive).

A pseudo-group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075700/p07570030.png" /> of transformations of a differentiable manifold is called a Lie pseudo-group of transformations defined by a system <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075700/p07570031.png" /> of partial differential equations if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075700/p07570032.png" /> consists of exactly those local transformations of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075700/p07570033.png" /> that satisfy the system <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075700/p07570034.png" />. 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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075700/p07570035.png" />-dimensional complex space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075700/p07570036.png" />.

b) The pseudo-group of all holomorphic local transformations of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075700/p07570037.png" /> with constant [[Jacobian|Jacobian]].

c) The pseudo-group of all holomorphic local transformations of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075700/p07570038.png" /> with Jacobian 1.

d) The Hamilton pseudo-group of all holomorphic local transformations of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075700/p07570039.png" /> (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075700/p07570040.png" /> even) preserving the differential <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075700/p07570041.png" />-form

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075700/p07570042.png" /></td> </tr></table>

e) The pseudo-group of all holomorphic local transformations of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075700/p07570043.png" /> preserving <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075700/p07570044.png" /> up to constant factor.

f) The contact pseudo-group of all holomorphic local transformations of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075700/p07570045.png" /> (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075700/p07570046.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075700/p07570047.png" />) preserving the differential <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075700/p07570048.png" />-form

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075700/p07570049.png" /></td> </tr></table>

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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075700/p07570050.png" /> of transformations of a manifold <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075700/p07570051.png" /> determines a pseudo-group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075700/p07570052.png" /> of transformations, consisting of the restrictions of the transformations from <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075700/p07570053.png" /> onto open subsets of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075700/p07570054.png" />. A pseudo-group of transformations of the form <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075700/p07570055.png" /> is called globalizable. E.g., a pseudo-group of local conformal transformations of the sphere <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075700/p07570056.png" /> is globalizable for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075700/p07570057.png" /> and not globalizable for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075700/p07570058.png" />.

A Lie pseudo-group of transformations is said to be of finite type if there is a natural number <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075700/p07570059.png" /> such that every local transformation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075700/p07570060.png" /> is uniquely determined by its <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075700/p07570061.png" />-jet at some point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075700/p07570062.png" />; the smallest such <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075700/p07570063.png" /> is called the degree, or type, of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075700/p07570064.png" />; if such a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075700/p07570065.png" /> does not exist, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075700/p07570066.png" /> 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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075700/p07570067.png" /> be a transitive Lie pseudo-group of transformations of an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075700/p07570068.png" />-dimensional manifold <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075700/p07570069.png" /> and let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075700/p07570070.png" /> be the family of all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075700/p07570071.png" />-jets of the local transformations in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075700/p07570072.png" /> that preserve a point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075700/p07570073.png" />, i.e. those <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075700/p07570074.png" /> for which <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075700/p07570075.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075700/p07570076.png" />. The set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075700/p07570077.png" />, endowed with the natural structure of a Lie group, is called the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075700/p07570079.png" />-th order isotropy group of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075700/p07570080.png" /> (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075700/p07570081.png" /> is also called the linear isotropy group of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075700/p07570082.png" />). The Lie algebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075700/p07570083.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075700/p07570084.png" /> can be naturally imbedded in the Lie algebra of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075700/p07570085.png" />-jets of vector fields on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075700/p07570086.png" /> at <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075700/p07570087.png" />. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075700/p07570088.png" /> is a Lie pseudo-group of transformations of order one, then the kernel <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075700/p07570089.png" /> of the natural homomorphism <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075700/p07570090.png" /> depends, for any <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075700/p07570091.png" />, only on the linear isotropy group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075700/p07570092.png" />, and is called its <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075700/p07570093.png" />-th extension. A Lie pseudo-group of transformations <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075700/p07570094.png" /> of order one is of finite type <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075700/p07570095.png" /> if and only if

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075700/p07570096.png" /></td> </tr></table>

If, moreover, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075700/p07570097.png" /> is irreducible, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075700/p07570098.png" /> (cf. ). A Lie pseudo-group of transformations <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075700/p07570099.png" /> 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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075700/p075700100.png" /> does not contain endomorphisms of rank 1 (cf. [[#References|[10]]]). Such linear Lie algebras are called elliptic.

One has calculated the Lie algebras of all extensions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075700/p075700101.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075700/p075700102.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075700/p075700103.png" /> is a Lie pseudo-group of transformations of order one, in terms of the linear isotropy algebra. More precisely, the Lie algebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075700/p075700104.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075700/p075700105.png" /> consists of the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075700/p075700106.png" />-jets of vector fields on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075700/p075700107.png" /> at <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075700/p075700108.png" /> having, in some local coordinate system <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075700/p075700109.png" />, the form

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075700/p075700110.png" /></td> </tr></table>

where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075700/p075700111.png" /> is an arbitrary tensor that is symmetric with respect to the lower indices and that satisfies the condition: For any fixed <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075700/p075700112.png" /> the matrix

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075700/p075700113.png" /></td> </tr></table>

belongs to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075700/p075700114.png" />, relative to some coordinate system <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075700/p075700115.png" />.

Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075700/p075700116.png" /> be an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075700/p075700117.png" />-dimensional differentiable manifold over the field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075700/p075700118.png" /> or <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075700/p075700119.png" />. Every transitive Lie pseudo-group of transformations <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075700/p075700120.png" /> of order <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075700/p075700121.png" /> on a manifold <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075700/p075700122.png" /> coincides with the pseudo-group of all local automorphism of some <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075700/p075700123.png" />-structure (cf. [[G-structure(2)|<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075700/p075700124.png" />-structure]]) of order <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075700/p075700125.png" /> on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075700/p075700126.png" /> (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. [[#References|[9]]]). The classification of these filtered Lie algebras can be given on the basis of the classification of simple graded Lie algebras (cf. [[#References|[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. [[#References|[8]]], [[#References|[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. , [[#References|[9]]]).

====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top"> S. Sternberg, "Lectures on differential geometry" , Prentice-Hall (1964) {{MR|0193578}} {{ZBL|0129.13102}} </TD></TR><TR><TD valign="top">[2a]</TD> <TD valign="top"> E. Cartan, "Sur la structure des groupes infinis de transformations" , ''Oeuvres complètes'' , '''2''' , Gauthier-Villars (1953) pp. 571–624 {{MR|1509054}} {{MR|1509040}} {{ZBL|36.0223.03}} {{ZBL|35.0176.04}} </TD></TR><TR><TD valign="top">[2b]</TD> <TD valign="top"> E. Cartan, "Sur la structure des groupes infinis de transformations" , ''Oeuvres complètes'' , '''2''' , Gauthier-Villars (1953) pp. 625–714 {{MR|1509054}} {{MR|1509040}} {{ZBL|36.0223.03}} {{ZBL|35.0176.04}} </TD></TR><TR><TD valign="top">[2c]</TD> <TD valign="top"> E. Cartan, "Les groupes de transformations continus, infinis, simples" , ''Oeuvres complètes'' , '''2''' , Gauthier-Villars (1953) pp. 857–925 {{MR|1509105}} {{ZBL|40.0193.02}} {{ZBL|38.0194.01}} </TD></TR><TR><TD valign="top">[2d]</TD> <TD valign="top"> E. Cartan, "Les groupes de transformations continus, infinis, simples" , ''Oeuvres complètes'' , '''2''' , Gauthier-Villars (1953) pp. 1335–1384 {{MR|1509105}} {{ZBL|40.0193.02}} {{ZBL|38.0194.01}} </TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top"> V. Guillemin, "Infinite dimensional primitive Lie algebras" ''J. Diff. Geom.'' , '''4''' : 3 (1970) pp. 257–282 {{MR|0268233}} {{ZBL|0223.17007}} </TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top"> S. Kobayashi, "Transformation groups in differential geometry" , Springer (1972) {{MR|0355886}} {{ZBL|0246.53031}} </TD></TR><TR><TD valign="top">[5a]</TD> <TD valign="top"> S. Kobayashi, T. Nagano, "On filtered Lie algebras and geometric structures I" ''J. Math. Mech.'' , '''13''' : 5 (1964) pp. 875–907 {{MR|0168704}} {{ZBL|0142.19504}} </TD></TR><TR><TD valign="top">[5b]</TD> <TD valign="top"> S. Kobayashi, T. Nagano, "On filtered Lie algebras and geometric structures III" ''J. Math. Mech.'' , '''14''' : 5 (1965) pp. 679–706 {{MR|0188364}} {{ZBL|}} </TD></TR><TR><TD valign="top">[6a]</TD> <TD valign="top"> M. Kuranishi, "On the local theory of continuous infinite pseudo groups I" ''Nagoya Math. J.'' , '''15''' (1959) pp. 225–260 {{MR|0116071}} {{ZBL|0212.56501}} </TD></TR><TR><TD valign="top">[6b]</TD> <TD valign="top"> M. Kuranishi, "On the local theory of continuous infinite pseudo groups II" ''Nagoya Math. J.'' , '''19''' (1961) pp. 55–91 {{MR|0142694}} {{ZBL|0212.56501}} </TD></TR><TR><TD valign="top">[7]</TD> <TD valign="top"> P. Libermann, "Pseudogroupes infinitésimaux attachées aux pseudogroupes de Lie" ''Bull. Soc. Math. France'' , '''87''' : 4 (1959) pp. 409–425 {{MR|123279}} {{ZBL|}} </TD></TR><TR><TD valign="top">[8]</TD> <TD valign="top"> S. Shnider, "The classification of real primitive infinite Lie algebras" ''J. Diff. Geom.'' , '''4''' : 1 (1970) pp. 81–89 {{MR|0285574}} {{ZBL|0244.17014}} </TD></TR><TR><TD valign="top">[9]</TD> <TD valign="top"> 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 {{MR|0217822}} {{ZBL|0277.58008}} </TD></TR><TR><TD valign="top">[10]</TD> <TD valign="top"> R.L. Wilson, "Irreducible Lie algebras of infinite type" ''Proc. Amer. Math. Soc.'' , '''29''' : 2 (1971) pp. 243–249 {{MR|0277582}} {{ZBL|0216.07401}} </TD></TR></table>



====Comments====


====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top"> C. Albert, P. Molino, "Pseudogroupes de Lie transitifs" , '''I–II''' , Hermann (1984–1987) {{MR|0904048}} {{MR|0770061}} {{ZBL|0682.53003}} {{ZBL|0563.53027}} </TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top"> J.F. Pommaret, "Systems of partial differential equations and Lie pseudogroups" , Gordon &amp; Breach (1978) {{MR|0517402}} {{ZBL|0418.35028}} {{ZBL|0401.58006}} </TD></TR></table>