| Algebraic structure →Group theory Group theory |
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Infinite dimensional Lie group
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Inmathematics, agroup action of a group on aset is agroup homomorphism from to some group (underfunction composition) of functions from to itself. It is said thatacts on.
Many sets oftransformations form agroup underfunction composition; for example, therotations around a point in the plane. It is often useful to consider the group as anabstract group, and to say that one has a group action of the abstract group that consists of performing the transformations of the group of transformations. The reason for distinguishing the group from the transformations is that, generally, a group of transformations of astructure acts also on various related structures; for example, the above rotation group also acts on triangles by transforming triangles into triangles.
If a group acts on a structure, it will usually also act on objects built from that structure. For example, the group ofEuclidean isometries acts onEuclidean space and also on the figures drawn in it; in particular, it acts on the set of alltriangles. Similarly, the group ofsymmetries of apolyhedron acts on thevertices, theedges, and thefaces of the polyhedron.
A group action on avector space is called arepresentation of the group. In the case of a finite-dimensional vector space, it allows one to identify many groups withsubgroups of thegeneral linear group, the group of theinvertible matrices ofdimension over afield.
Thesymmetric group acts on anyset with elements by permuting the elements of the set. Although the group of allpermutations of a set depends formally on the set, the concept of group action allows one to consider a single group for studying the permutations of all sets with the samecardinality.
If is agroup withidentity element, and is a set, then a (left)group action of onX is afunction
that satisfies the following twoaxioms:[1]
| Identity: | |
| Compatibility: |
for allg andh inG and allx in.
The group is then said to act on (from the left). A set together with an action of is called a (left)-set.
It can be notationally convenient tocurry the action, so that, instead, one has a collection oftransformationsαg :X →X, with one transformationαg for each group elementg ∈G. The identity and compatibility relations then read
and
The second axiom states that the function composition is compatible with the group multiplication; they form acommutative diagram. This axiom can be shortened even further, and written as.
With the above understanding, it is very common to avoid writing entirely, and to replace it with either a dot, or with nothing at all. Thus,α(g,x) can be shortened tog⋅x orgx, especially when the action is clear from context. The axioms are then
From these two axioms, it follows that for any fixedg in, the function fromX to itself which mapsx tog⋅x is abijection, with inverse bijection the corresponding map forg−1. Therefore, one may equivalently define a group action ofG onX as a group homomorphism fromG into the symmetric groupSym(X) of all bijections fromX to itself.[2]
Likewise, aright group action of on is a function
that satisfies the analogous axioms:[3]
| Identity: | |
| Compatibility: |
(withα(x,g) often shortened toxg orx⋅g when the action being considered is clear from context)
| Identity: | |
| Compatibility: |
for allg andh inG and allx inX.
The difference between left and right actions is in the order in which a productgh acts onx. For a left action,h acts first, followed byg second. For a right action,g acts first, followed byh second. Because of the formula(gh)−1 =h−1g−1, a left action can be constructed from a right action by composing with the inverse operation of the group. Also, a right action of a groupG onX can be considered as a left action of itsopposite groupGop onX.
Thus, for establishing general properties of group actions, it suffices to consider only left actions. However, there are cases where this is not possible. For example, the multiplication of a groupinduces both a left action and a right action on the group itself—multiplication on the left and on the right, respectively.
LetG be a group acting on a setX. The action is calledfaithful oreffective ifg⋅x =x for allx ∈X implies thatg =eG. Equivalently, thehomomorphism fromG to the group of bijections ofX corresponding to the action isinjective.
The action is calledfree (orsemiregular orfixed-point free) if the statement thatg⋅x =x for somex ∈X already implies thatg =eG. In other words, no non-trivial element ofG fixes a point ofX. This is a much stronger property than faithfulness.
For example, the action of any group on itself by left multiplication is free. This observation impliesCayley's theorem that any group can beembedded in a symmetric group (which is infinite when the group is). A finite group may act faithfully on a set of size much smaller than its cardinality (however such an action cannot be free). For instance the abelian 2-group(Z / 2Z)n (of cardinality2n) acts faithfully on a set of size2n. This is not always the case, for example thecyclic groupZ / 2nZ cannot act faithfully on a set of size less than2n.
In general the smallest set on which a faithful action can be defined can vary greatly for groups of the same size. For example, three groups of size 120 are the symmetric groupS5, the icosahedral groupA5 ×Z / 2Z and the cyclic groupZ / 120Z. The smallest sets on which faithful actions can be defined for these groups are of size 5, 7, and 16 respectively.
The action ofG onX is calledtransitive if for any two pointsx,y ∈X there exists ag ∈G so thatg ⋅x =y.
The action issimply transitive (orsharply transitive, orregular) if it is both transitive and free. This means that givenx,y ∈X there is exactly oneg ∈G such thatg ⋅x =y. IfX is acted upon simply transitively by a groupG then it is called aprincipal homogeneous space forG or aG-torsor.
For an integern ≥ 1, the action isn-transitive ifX has at leastn elements, and for any pair ofn-tuples(x1, ...,xn), (y1, ...,yn) ∈Xn with pairwise distinct entries (that isxi ≠xj,yi ≠yj wheni ≠j) there exists ag ∈G such thatg⋅xi =yi fori = 1, ...,n. In other words, the action on the subset ofXn of tuples without repeated entries is transitive. Forn = 2, 3 this is often called double, respectively triple, transitivity. The class of2-transitive groups (that is, subgroups of a finite symmetric group whose action is 2-transitive) and more generallymultiply transitive groups is well-studied in finite group theory.
An action issharplyn-transitive when the action on tuples without repeated entries inXn is sharply transitive.
The action of the symmetric group ofX is transitive, in factn-transitive for anyn up to the cardinality ofX. IfX has cardinalityn, the action of thealternating group is(n − 2)-transitive but not(n − 1)-transitive.
The action of thegeneral linear group of a vector spaceV on the setV ∖ {0} of non-zero vectors is transitive, but not 2-transitive (similarly for the action of thespecial linear group if the dimension ofv is at least 2). The action of theorthogonal group of a Euclidean space is not transitive on nonzero vectors but it is on theunit sphere.
The action ofG onX is calledprimitive if there is nopartition ofX preserved by all elements ofG apart from the trivial partitions (the partition in a single piece and itsdual, the partition intosingletons).
Assume that is atopological space and the action of is byhomeomorphisms.
The action iswandering if every has aneighbourhood such that there are only finitely many with.[4]
More generally, a point is called a point of discontinuity for the action of if there is an open subset such that there are only finitely many with. Thedomain of discontinuity of the action is the set of all points of discontinuity. Equivalently it is the largest-stable open subset such that the action of on is wandering.[5] In a dynamical context this is also called awandering set.
The action isproperly discontinuous if for everycompact subset there are only finitely many such that. This is strictly stronger than wandering; for instance the action of on given by is wandering and free but not properly discontinuous.[6]
The action bydeck transformations of thefundamental group of a locallysimply connected space on auniversal cover is wandering and free. Such actions can be characterized by the following property: every has a neighbourhood such that for every.[7] Actions with this property are sometimes calledfreely discontinuous, and the largest subset on which the action is freely discontinuous is then called thefree regular set.[8]
An action of a group on alocally compact space is calledcocompact if there exists a compact subset such that. For a properly discontinuous action, cocompactness is equivalent to compactness of thequotient space.
Now assumeG is atopological group andX a topological space on which it acts by homeomorphisms. The action is said to becontinuous if the mapG ×X →X is continuous for theproduct topology.
The action is said to beproper if the mapG ×X →X ×X defined by(g,x) ↦ (x,g⋅x) isproper.[9] This means that given compact setsK,K′ the set ofg ∈G such thatg⋅K ∩K′ ≠ ∅ is compact. In particular, this is equivalent to proper discontinuity ifG is adiscrete group.
It is said to belocally free if there exists a neighbourhoodU ofeG such thatg⋅x ≠x for allx ∈X andg ∈U ∖ {eG}.
The action is said to bestrongly continuous if the orbital mapg ↦g⋅x is continuous for everyx ∈X. Contrary to what the name suggests, this is a weaker property than continuity of the action.[citation needed]
IfG is aLie group andX adifferentiable manifold, then the subspace ofsmooth points for the action is the set of pointsx ∈X such that the mapg ↦g⋅x issmooth. There is a well-developed theory ofLie group actions, i.e. action which are smooth on the whole space.
Ifg acts bylinear transformations on amodule over acommutative ring, the action is said to beirreducible if there are no proper nonzerog-invariant submodules. It is said to besemisimple if it decomposes as adirect sum of irreducible actions.
Consider a groupG acting on a setX. Theorbit of an elementx inX is the set of elements inX to whichx can be moved by the elements ofG. The orbit ofx is denoted byG⋅x:
The defining properties of a group guarantee that the set of orbits of (pointsx in)X under the action ofG form apartition ofX. The associatedequivalence relation is defined by sayingx ~yif and only if there exists ag inG withg⋅x =y. The orbits are then theequivalence classes under this relation; two elementsx andy are equivalent if and only if their orbits are the same, that is,G⋅x =G⋅y.
The group action istransitive if and only if it has exactly one orbit, that is, if there existsx inX withG⋅x =X. This is the case if and only ifG⋅x =X forallx inX (given thatX is non-empty).
The set of all orbits ofX under the action ofG is written asX /G (or, less frequently, asG \X), and is called thequotient of the action. In geometric situations it may be called theorbit space, while in algebraic situations it may be called the space ofcoinvariants, and writtenXG, by contrast with the invariants (fixed points), denotedXG: the coinvariants are aquotient while the invariants are asubset. The coinvariant terminology and notation are used particularly ingroup cohomology andgroup homology, which use the same superscript/subscript convention.
IfY is asubset ofX, thenG⋅Y denotes the set{g⋅y :g ∈G andy ∈Y}. The subsetY is said to beinvariant underG ifG⋅Y =Y (which is equivalentG⋅Y ⊆Y). In that case,G also operates onY byrestricting the action toY. The subsetY is calledfixed underG ifg⋅y =y for allg inG and ally inY. Every subset that is fixed underG is also invariant underG, but not conversely.
Every orbit is an invariant subset ofX on whichG actstransitively. Conversely, any invariant subset ofX is a union of orbits. The action ofG onX istransitive if and only if all elements are equivalent, meaning that there is only one orbit.
AG-invariant element ofX isx ∈X such thatg⋅x =x for allg ∈G. The set of all suchx is denotedXG and called theG-invariants ofX. WhenX is aG-module,XG is the zerothcohomology group ofG with coefficients inX, and the higher cohomology groups are thederived functors of thefunctor ofG-invariants.
Giveng inG andx inX withg⋅x =x, it is said that "x is a fixed point ofg" or that "g fixesx". For everyx inX, thestabilizer subgroup ofG with respect tox (also called theisotropy group orlittle group[10]) is the set of all elements inG that fixx:This is asubgroup ofG, though typically not a normal one. The action ofG onX isfree if and only if all stabilizers are trivial. The kernelN of the homomorphism with the symmetric group,G → Sym(X), is given by theintersection of the stabilizersGx for allx inX. IfN is trivial, the action is said to be faithful (or effective).
Letx andy be two elements inX, and letg be a group element such thaty =g⋅x. Then the two stabilizer groupsGx andGy are related byGy =gGxg−1.
Proof: by definition,h ∈Gy if and only ifh⋅(g⋅x) =g⋅x. Applyingg−1 to both sides of this equality yields(g−1hg)⋅x =x; that is,g−1hg ∈Gx.
An opposite inclusion follows similarly by takingh ∈Gx andx =g−1⋅y.
The above says that the stabilizers of elements in the same orbit areconjugate to each other. Thus, to each orbit, we can associate aconjugacy class of a subgroup ofG (that is, the set of all conjugates of the subgroup). Let(H) denote the conjugacy class ofH. Then the orbitO has type(H) if the stabilizerGx of some/anyx inO belongs to(H). A maximal orbit type is often called aprincipal orbit type.
Orbits and stabilizers are closely related. For a fixedx inX, consider the mapf :G →X given byg ↦g⋅x. By definition the imagef(G) of this map is the orbitG⋅x. The condition for two elements to have the same image isIn other words,f(g) =f(h)if and only ifg andh lie in the samecoset for the stabilizer subgroupGx. Thus, thefiberf−1({y}) off over anyy inG⋅x is contained in such a coset, and every such coset also occurs as a fiber. Thereforef induces abijection between the setG /Gx of cosets for the stabilizer subgroup and the orbitG⋅x, which sendsgGx ↦g⋅x.[11] This result is known as theorbit–stabilizer theorem.
IfG is finite then the orbit–stabilizer theorem, together withLagrange's theorem, givesIn other words, the length of the orbit ofx times the order of its stabilizer is theorder of the group. In particular that implies that the orbit length is a divisor of the group order.
This result is especially useful since it can be employed for counting arguments (typically in situations whereX is finite as well).
A result closely related to the orbit–stabilizer theorem isBurnside's lemma:whereXg is the set of points fixed byg. This result is mainly of use whenG andX are finite, when it can be interpreted as follows: the number of orbits is equal to the average number of points fixed per group element.
Fixing a groupG, the set of formal differences of finiteG-sets forms a ring called theBurnside ring ofG, where addition corresponds todisjoint union, and multiplication toCartesian product.
The notion of group action can be encoded by theactiongroupoidG′ =G ⋉X associated to the group action. The stabilizers of the action are the vertex groups of the groupoid and the orbits of the action are its components.
IfX andY are twoG-sets, amorphism fromX toY is a functionf :X →Y such thatf(g⋅x) =g⋅f(x) for allg inG and allx inX. Morphisms ofG-sets are also calledequivariant maps orG-maps.
The composition of two morphisms is again a morphism. If a morphismf is bijective, then its inverse is also a morphism. In this casef is called anisomorphism, and the twoG-setsX andY are calledisomorphic; for all practical purposes, isomorphicG-sets are indistinguishable.
Some example isomorphisms:
With this notion of morphism, the collection of allG-sets forms acategory; this category is aGrothendieck topos (in fact, assuming a classicalmetalogic, thistopos will even be Boolean).
We can also consider actions ofmonoids on sets, by using the same two axioms as above. This does not define bijective maps and equivalence relations however. Seesemigroup action.
Instead of actions on sets, we can define actions of groups and monoids on objects of an arbitrary category: start with an objectX of some category, and then define an action onX as a monoid homomorphism into the monoid ofendomorphisms ofX. IfX has an underlying set, then all definitions and facts stated above can be carried over. For example, if we take the category of vector spaces, we obtaingroup representations in this fashion.
We can view a groupG as a category with a single object in which every morphism isinvertible.[15] A (left) group action is then nothing but a (covariant)functor fromG to thecategory of sets, and a group representation is a functor fromG to thecategory of vector spaces.[16] A morphism betweenG-sets is then anatural transformation between the group action functors.[17] In analogy, an action of agroupoid is a functor from the groupoid to the category of sets or to some other category.
In addition tocontinuous actions of topological groups on topological spaces, one also often considerssmooth actions of Lie groups onsmooth manifolds, regular actions ofalgebraic groups onalgebraic varieties, andactions ofgroup schemes onschemes. All of these are examples ofgroup objects acting on objects of their respective category.
