# Pseudo-group

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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. ) 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. ). 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. ). The classification of these filtered Lie algebras can be given on the basis of the classification of simple graded Lie algebras (cf. ). 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. , ). 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. , ).

How to Cite This Entry:
Pseudo-group. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Pseudo-group&oldid=48346
This article was adapted from an original article by E.B. Vinberg (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article