Representations of the group of the quaternions

In Mathématiques the representations of the Groupe of the Quaternion S are an example of application of the theory of the Représentations of a group finished. It illustrates the Théorème of Artin-Wedderburn and highlights a left body (id not Commutatif) containing that of the real numbers.

On the body of the complex numbers, there exist five irreducible representations that the characters highlight quickly.

On the body of the real numbers, there exist also five irreducible representations of which one has a property specific to the semi-simple algebras. The morphisms of the subalgebra simple partner form a left body containing that of realities. The traditional configuration of the Lemme of Schur where the endomorphisms commutating with an irreducible representation are limited to the Homothétie S is not checked here.

This property, making it possible to build left bodies, using a associative Algèbre is one of the fundamental methods of the theory of the bodies. The term in the past used was Nombre hypercomplexe.

Group of the quaternions analyzes

Definition and properties of the group

See also: Group of quaternions

There exist many manners of defining the group of the quaternions. That used here is the abstracted definition, starting from two generators. The group is noted Q 8, it is noted multiplicativement, the neutral element is noted 1 and the two generators has and B . They check the following equalities:

a^4=b^4=1 \; , \ quad bab^ {- 1} =a^ {- 1} \; , \ quad a^2=b^2

Its table is the following one:

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Element -1 commutates with all the others, the group {1, -1} is thus a isomorphous Sous-groupe distinguished with the cyclic Groupe of a nature two, noted here C 2. The quotient of Q 8 by the group C 2 gives a group to four elements. One recognizes the Groupe of Klein noted here V .

There exists in fact five groups of order eight. Three are abelian: the group of Klein with 8 elements C 23, the group Produces direct cyclic group of order four by that of order two C 2x C 4 and the group cylic of order eight C 8. There are two nonabelian, the diédral group of a semi-direct nature four Produit cyclic group of order four by that of order two and Q 8 produces semi-direct V and of C 2.

There from of deduced the existence from five classes of conjugation, 1 = {1}, -1 = {- 1}, has = { has , - is }, B = { B , - B } and AB = { ab , - ab }, these notations will be used throughout the article.

Representation on the body of the complexes

Irreducible characters

See also: Character of a representation of a group finished

The quotient of Q 8 by C 2 is an isomorphous abelian group with V . This remark determines four irreducible characters correspondent with representations of degree a . Indeed, if χ indicates a character of degree a of V and φ the surjective morphism canonical of Q 8 in V then χ O φ is a representation of degree a of Q 8 and thus an irreducible character. One notes χ1 here the character of the commonplace representation and χX the character equal to -1 on X indicating here one of the classes of conjugation has , B or AB and 1 elsewhere.

The number of irreducible representations to an isomorphism close is equal to the number of classes of conjugation. There thus remains only one single irreducible representation ρ to be found and its associated character, noted here χρ. The image of element 1 of the group by χρ is equal to two and represents the degree of the representation ρ. To be convinced some it is enough to notice that the sum of the squares of the degrees of the various irreducible representations of the group is equal to its order. χρ is of standard equal to 1 and is orthogonal with the other irreducible characters. These considerations determine the last character completely. Let us summarize these results in a table:

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Irreducible representation of degree two

See also: induced Representation of a group finished

A fast method to determine the last representation is given by the induced representations. To obtain directly a representation of degree two, it is necessary to consider a sub-group H of order four . Let us define H as the isomorphous group with C 4 generated by has . A representation θ must then be selected to be then induced on Q 8. This representation θ must be selected in such manner so that the induced representation ρ is of character orthogonal with the representations of degree 1. The law of Réciprocité of Frobenius simply makes it possible to check this orthogonality. It is expressed in the following way:

_ {Q_8} =< \ chi_ {\ theta} \, |\, Res_H^ {Q_8} \ chi_X>_H
On 1 and -1 all the cararactères of the representations of degree 1 are equal to 1. On has and - has , they are worth either 1 or -1. If one chooses for character that which with has associates I , then required orthogonalities are well obtained. The use of the Criterion of irreducibility of Mackey and the remark that all the interior automorphisms leave the elements of the invariant group make it possible to conclude without calculation. And the representation ρ is well the induced representation of θ.

To know this representation, it is enough to determine the matrices has and of B because they are generating group. If M a indicates the matrix of has and M b that of B , one obtains:

M_a = \ begin {pmatrix} I & 0 \ \ 0 & - I \ end {pmatrix} \ quad and \ quad M_b = \ begin {pmatrix} 0 & -1 \ \ 1 & 0 \ end {pmatrix}

Semi-simple algebra

See also: semi-simple Algebra

In the case of the complexes, it is noticed that the four matrices M 1, M a, M b, M ab form a base of the matrices 2x2. Indeed:

M_1 = \ begin {pmatrix} 1 & 0 \ \ 0 & 1 \ end {pmatrix} \; , \ quad M_a = \ begin {pmatrix} I & 0 \ \ 0 & - I \ end {pmatrix} \; , \ quad M_b = \ begin {pmatrix} 0 & -1 \ \ 1 & 0 \ end {pmatrix} \; and \ quad M_ {ab} = \ begin {pmatrix} 0 & - I \ \ - I & 0 \ end {pmatrix}
If M c indicates the matrix of the element C of C , the algebra C checks the equality:
\ forall C \ in \ mathbb C \ quad \ exists! (a_i) _ {I \ in} \ in \ mathbb C^8 \ quad M_c = \ begin {pmatrix} a_1 & 0 & 0 & 0 & 0 & 0 \ \ 0 & a_2 & 0 & 0 & 0 & 0 \ \ 0 & 0 & a_3 & 0 & 0 & 0 \ \ 0 & 0 & 0 & a_4 & 0 & 0 \ \ 0 & 0 & 0 & 0 & a_5 & a_7 \ \ 0 & 0 & 0 & 0 & a_6 & a_8 \ end {pmatrix}
Its center is quite equal to C 5 and is isomorphous with the whole of the central functions.

Representations on the body of realities

Dimension of the irreducible representations

See also: Lemma of Schur

On the body of realities, the situation is more delicate. It is easy to check that no basic change allows the matrices M a and M b to be with real coefficients at the same time. The irreducible representation complexes dimension two is thus not real.

The assumptions which make it possible the Lemme of Schur to guarantee that the morphisms of subalgebras are the Homothétie S are not gathered, because the body R of realities is not algebraically closed. It states just that the morphisms correspond to isomorphisms. The theory of the characters is inoperative in this context.

On the other hand, this conclusion is sufficient to apply the Théorème of Maschke. The algebra as the Q 8-module are thus semi-simple. One from of deduced that R is direct sum of simple subalgebras.

Four of these subalgebras are known, they correspond to the representations of degree a , which all are real. There exists an additional S of the direct sum of the four representations of degree a stable . I.e. having the properties of a subalgebra. We know that S is of dimension four . As no subalgebra is isomorphous with another S any factor of dimension a does not comprise. As any subalgebra of real size two is also a complex subalgebra, and that there is only one, S does not comprise a subalgebra of dimension two. There cannot be either subalgebra of dimension three because he would admit additional stable, which would be a subalgebra of dimension a . In conclusion:

* the algebra R '' contains five simple subalgebras, four of dimension a and one of size four .

Irreducible representation of degree four

See also: Produces tensorial and representations of finished groups

That is to say V the subalgebra of the irreducible complex representation of degree two. It is also a real representation of degree four, this remark is a direct consequence of a tensorial equality:

V_ {\ mathbb R} = \ mathbb C \ otimes_ {\ mathbb R} V \;
In term matric, if R x indicates the real matrix of M x where X indicates an element of the group of the quaternions, one obtains the three following equalities:
R_a = \ begin {pmatrix} 0 & -1 & 0 & 0 \ \ 1 & 0 & 0 & 0 \ \ 0 & 0 & 0 & 1 \ \ 0 & 0 & -1 & 0 \ end {pmatrix} \ quad, \ quad R_b = \ begin {pmatrix} 0 & 0 & -1 & 0 \ \ 0 & 0 & 0 & -1 \ \ 1 & 0 & 0 & 0 \ \ 0 & 1 & 0 & 0 \ end {pmatrix} \ quad and \ quad R_ {ab} = \ begin {pmatrix} 0 & 0 & 0 & -1 \ \ 0 & 0 & -1 & 0 \ \ 0 & 1 & 0 & 0 \ \ -1 & 0 & 0 & 0 \ end {pmatrix}
The character of has as a complex representation is null, the representation corresponds to a direct sum of two copies of ρ. It is thus quite irreducible in R .

It is checked that the family (1, has , B , ab ) is a base B of V R. Let us note σ the representation associated and H x the matrix with σx where X is an element of R in the base B . One obtains a system of matrices very close to the precedent:

H_a = \ begin {pmatrix} 0 & -1 & 0 & 0 \ \ 1 & 0 & 0 & 0 \ \ 0 & 0 & 0 & -1 \ \ 0 & 0 & 1 & 0 \ end {pmatrix} \ quad, \ quad H_b = \ begin {pmatrix} 0 & 0 & -1 & 0 \ \ 0 & 0 & 0 & 1 \ \ 1 & 0 & 0 & 0 \ \ 0 & -1 & 0 & 0 \ end {pmatrix} \ quad and \ quad H_ {ab} = \ begin {pmatrix} 0 & 0 & 0 & -1 \ \ 0 & 0 & -1 & 0 \ \ 0 & 1 & 0 & 0 \ \ 1 & 0 & 0 & 0 \ end {pmatrix}

One from of deduced the generic matrix from an element of R :

\ forall H \ in \ mathbb R R} \ quad \ exists! (h_i) _ {I \ in} \ in \ mathbb R^4 \ quad/\ quad H_h = h_1. 1 + h_2.a + h_3.b+h_4.ab= \ begin {pmatrix} h_1 & - h_2 & - h_3 & - h_4 \ \ h_2 & h_1 & - h_4 & h_3 \ \ h_3 & h_4 & h_1 & - h_2 \ \ h_4 & - h_3 & h_2 & h_1 \ end {pmatrix}

Morphisms of the representation

See also: Theorem of Artin-Wedderburn

Let us seek the unit D morphisms of the Q 8-module V R, i.e. the whole of the applications D vérifant:

\ forall v_1, v_2 \ in V_ {\ mathbb R} \ quad \ forall h_1, h_2 \ in \ mathbb R R} \ quad D (h_1.v_1 + h_2.v_2) =h_1.d (v_1) + h_2.d (v_2) \;

Here, there exists an isomorphism of module between R and V R, one from of deduced that R is a simple ring, the morphisms are thus easy to determine, if D 1 indicates the value D (1) one obtains:

\ forall D \ in \ mathbb D \ quad \ forall v \ in \ mathbb R R} \ quad D (v) = D (v.1) =d (v) .d_1 \;

If φ indicates the application of D in R which associates with D the element D 1, then the following property is obtained:

\ forall d^1, d^2 \ in \ mathbb D \ quad \ varphi (d^1 \ circ d^2) = \ varphi (d^2). \ varphi (d^1) \ quad because \ quad d^1 \ circ d^2 (1) = d^1 (d^2 (1))= d^2 (1) .d^1 (1) \;
If D op indicates the opposite ring of D , i.e. that where the multiplication * is defined by the following equality, there exists an isomorphism of ring between D op and R :
\ forall d^1, d^2 \ in \ mathbb D \ quad d^1*d^2=d^2 \ circ d^1 \ quad and \ quad \ mathbb D^ {COp} = Hom^ {Q_8} _ {\ mathbb R} (V_ {\ mathbb R}) ^ {COp} \ simeq \ mathbb R R} \;

The lemma of Schur applies, even if the assumptions indicating that the morphisms are homotheties are not present, it states nevertheless that D * is made up of isomorphisms. What shows that D is a left body. It is known:

* the left body D is named body of the Quaternion S.
This method is now the principal one to build the left bodies and represents a branch of the theory of the bodies. Initially, such a body was regarded as a whole of numbers hypercomplexes. One now rather speaks about associative Algèbre noncommutative.

Checking by calculation

Let us determine the matric representation of D . For that let us note D δ where δ is an element of V R the matrix of the element D of D such as φ ( D ) = δ . Any element v of V R, D associates vδ . If the coordinates of δ in the base B are (δi) for I varying from 1 to 4, one obtains:

D_ {\ delta} = \ begin {pmatrix} \ delta_1 & - \ delta_2 & - \ delta_3 & - \ delta_4 \ \ \ delta_2 & \ delta_1 & \ delta_4 & - \ delta_3 \ \ \ delta_3 & - \ delta_4 & \ delta_1 & \ delta_2 \ \ \ delta_4 & \ delta_3 & - \ delta_2 & \ delta_1 \ end {pmatrix}

One then checks the lemma of Schur in a traditional form of commutation of matrices:

\ forall \ delta, H \ in {\ mathbb R} ^4 \ quad D_ {\ delta}. H_h=H_h.D_ {\ delta} \;

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