# 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]).

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