Semi-simple algebra

In Mathématiques and more particularly in Algèbre, algebra L Has, where has indicates a ring, is described of semi-simple or completely reducible if and only if the structure of ring associated with L is

It is present in many mathematical branches, one can quote the Linear algebra the Arithmétique, the Théorie of the representations of a group finished that of the groups of Dregs or that of the algebras of Dregs. It, for example, is used to show the criterion of Réciprocité of Frobenius.

The theory of the semi-simple algebras , is based on the Lemme of Schur and the Théorème of Artin-Wedderburn.

Definitions

Throughout this article, the following notations are used: has indicates a unit ring not necessarily Commutatif, L a unit algebra on the ring has . K indicates a body, not necessarily commutative and E a vector Space.

Several definitions are necessary to the comprehension of the concept of algebra semi-simple. The definitions differ a little from the case of the semi-simple module. For example a module is known as simple if he does not admit an other submodule only itself and the null unit. The archetypal situation of the simple algebra is that of L ( E ), the whole of the Endomorphisme S of E . He admits many subalgebras, for example if p is a projector, then the whole of the endomorphisms of the form p O has where has described the endomorphisms is a subalgebra, one uses the following definition then:

* the algebra L is known as simple if and only if only the ideal bilatères is the null unit and L itself.

K is a simple algebra regarded as a vector space on itself. The ring of the whole Z is not a Z - algebra simple, indeed any submodule N . Z if N is an entirety contains the submodule 2.nZ . This definition spreads with the modules, a module is known as simple if he admits like submodule only itself and the null unit.

* Is F an ideal of L , F is also called a factor invariant .

The body of the complex numbers is an algebra as a vector space real, R the whole of the real numbers is a simple submodule but is not a factor invariant within the meaning of the algebra. If E is of dimension two basic ( E ₁, E ₂) and if L is the K algebra of the endomorphisms having the base like clean vectors, then the subset of L having for image of E ₁ the null vector is a factor invariant.

* Is F a factor invariant, F is known as direct factor if and only if there exists an additional invariant by F .

Before giving the definition of a semi-simple algebra let us recall that a module is known as semi-simple if and only if it is direct Somme of factors invariants. The following notation is used in the article _L L indicates the module L on the ring L .

* an algebra L is known as semi-simple if and only if the modules _L L on the right and on the left are it.

This definition immediately induces the definition of a semi-simple ring:

* a ring is known as semi-simple if and only if it is semi-simple as an algebra on itself.

History

Origins

The history of the study of the concept of algebra is initially closely related to that of the relation between the Linear algebra and the Théorie of the groups. Arthur Cayley (1821 - 1895) develops in 1850 the concept of matrix. This concept brings many services of which one is at the origin of the concept. It makes it possible to incarnate groups and particularly the groups of Welshman and to study them under a new axis, that of a group of matrices. In the beginning only the finished case is studied, the analysis highlights a new structure, which one now regards as the algebra Endomorphisme S generated by the Automorphisme S of the group.

Camille Jordan (1838 - 1922) , the large specialist in the time with Cayley, uses it intensively. In 1869, it enables him to show the existence of a chain of composition for the groups finished known under the name of Théorème of Jordan-Hölder. The character of unicity of such a chain is shown by Otto Hölder (1859 - 1937) twenty years later. This theorem has two possible readings, one on the groups finished, the other on the modules. The second reading corresponds to an essential property structural, it is one of the origins of the interest for what will become a branch of mathematics, the commutative Algèbre. The analysis of the Groupe of Welshman also offers prospects in linear algebra. It leads Jordan to study the endomorphisms in Dimension finished through this algebra and allows a comprehension as major as final of their structure. This result, is published in a book of synthesis in 1870. It is known under the name of Réduction of Jordan and applies to the finished Corps first, i.e. the bodies of the entireties modulo a prime number.

The consequences of work of Jordan are considerable, its book of synthesis becomes the reference on the theories of the groups, Welshman and the linear algebra. It shows on the one hand that the analysis of the groups through the linear Groupe is a fertile step, and in addition that the structure of algebra is rich in lesson at the same time in terms of module and linear algebra.

Theory of the groups

The research of the comprehension of the groups becomes a major subject in mathematics. Across their own interest, the comprehension of this structure is the key many subjects. The Theory of Welsh the place in the middle of the problem of the algebraic equation and the consequences are multiple, the analysis of the structures of body is identified at that time with the theory of Welshman and the comprehension of many rings, useful in Arithmétique is based on this theory. The geometry quickly does not make any more exception. In 1870, two mathematicians Felix Klein (1849 1925) and Sophus Lie (1842 1899) visit Jordan with Paris. They are particularly interested by one of its old woman publications one year old on the analysis of one geometry using a group of symmetries. Sophus Lie develops a theory of the continuous groups and Klein, in its famous program classifies the geometries through the groups. They lose their character primarily finished.

Georg Frobenius (1849 - 1917) , following a correspondence with Richard Dedekind (1831 1916) is interested in the finished groups and particularly in the concept of factorization of a matric representation called at the time determinant of group and now fallen in disuse. These letters are at the origin of the theory of the representations of a group. In 1897 it seizes the proximity between a representation i.e. a group which operates linearly on a vector space and a module, where a ring operates on space. The jump is crossed, the group is linearized and becomes a module. Any progress on the modules having a structure equivalent to that of a module on a group is prone to make progress the theory of the representation and thus that of the groups.

Heinrich Maschke (1853 1908) , a pupil of Klein, is the first to show the theorem which determines the structuring element of this type of module, it is semi-simple. It has strong analogies with the Euclidean rings like that of the whole numbers. They break up into a series of simple modules which correspond a little to the prime numbers, with the difference that there is only one finished number.

Structure of algebra

A structure appears increasingly central, that of semi-simple Algèbre. In the case of the representations, it corresponds to make operate the linear extension of the group, no longer on an unspecified vector space, but on itself. Other mathematical branches bring naturally to the use of this concept. A extension galoisienne has a similar structure and the theory of the bodies supposes the study of these objects. Lastly, the continuous groups developed by Lie lay out in each point of a tangent Espace equipped with a semi-simple structure of algebra. At the dawn of the 20th century, this subject becomes major, of the mathematicians of different background study this concept. The theorem of decomposition of the modules apply because an algebra also has a structure of module.

William Burnside (1852 1927) quickly seizes the range of the approach of Frobenius. The importance of the structure of the algebra subjacent with a linear group does not escape to him. It establishes as of 1897 in the first edition of its book of reference on the group finished a first result. If the body is algebraically closed the whole of the Endomorphisme S of a vector space of finished size is a simple algebra. An example of building blocs is then clarified.

Leonard Dickson (1874 - 1954) by the way writes in 1896 its thesis of doctorate the groups of Welshman as of the linear groups on unspecified finished bodies, thus generalizing the results of Jordan. It shows that any commutative finished body is a Extension of Welshman of a body first. It is published in Europe in 1901. The basic structure is that of a semi-simple algebra. If the Welshman approach allows only the study of the commutative bodies, the semi-simple algebras allow also that of the left bodies (IE. noncommutative), Dickson develops the theory general of the bodies and finds many examples of left bodies. From this period two theories: that of Welshman and that of the bodies begin their separation.

Elie Cartan (1869 - 1951) is interested as of its thèsequ' it supports in 1894 with the algebras of Dregs. All structures of the simple and semi-simple algebras on the complexes are treated there. With Joseph Wedderburn (1882 - 1948) , it studies the generic structure of these algebras. Cartan clarifies the structure of the semi-simple algebras for the case of the complex numbers. In 1907 Wedderburn publishes its article perhaps most famous. It generalizes the results of Cartan for the algebras on an unspecified body which one calls at the time the numbers hypercomplexes. This generalization is important, because all the examples of applications quoted previously use left bodies.

Structure of ring

The theorem of Wedderburn modifies the situation, it exists natural bodies for any simple algebra, even if these bodies are a priori noncommutative. The theorem must thus be able to be expressed in term of ring. If Wedderburn does not do it, in 1908 it proposes nevertheless a classification comprising on the one hand the rings with radicals and on the other hand thesimple ones. This decomposition becomes the base of the theory of the rings for the half-century to come. Emmy Noether is often regarded as the mother of the modern theory of the rings.

A great figure of research in this field is Emmy Noether (1882 - 1935) . It develops the theory of the noncommutative rings and founds a general theory of the ideals. The concept of irreducible ideal, corresponding to the simple algebra, is developed, as well as the theory of the rings of which any strictly increasing ascending chain of ideals is finished. These rings is now named in its honor.

Emil Artin (1898 - 1962) particularly studies a case whose study is initiated by Noether, that of the rings of which any strictly decreasing downward chain of ideals is finished. A semi-simple ring of length finished is a ring artinien and noethérien. In 1927 it finds the form final of the theorem. Without its linear formalism, the theorem takes its maximum range, it becomes an important result of the noncommutative algebra. A broad class of rings, is isomorphous with a product of algebras associative on unspecified bodies.

If the theorem is final, one of attributes remains open. Which class of rings other than those at the same time artiniens and noethériens satisfy the theorem? A first brief reply is brought in 1939. C. Hopkins shows that, only the condition on the downward chain is necessary. The true opening is nevertheless the work of NR. jacobson, which finds the condition. It relates to the concept of radical, now essential for the study of the semi-simple rings.

Examples

Endomorphisms of E

* the whole of the endomorphisms L ( E ) is a simple algebra if E is a finished vector space of dimension.

This result is known under the name of theorem of Burnside , it is shown in the article Théorème of Artin-Wedderburn.

Algebra of group

Artinien ring

Properties inherited the modules

Characterization of a semi-simple algebra

* the three following proposals is equivalent:

(I) L is a semi-simple algebra.

(II) L is nap of simple subalgebras.

(III) Any subalgebra is a direct factor.

The consequences also apply to the algebras:

* Supposons that L is not semi-simple, then the sum of all the simple subalgebras of L is a semi-simple subalgebra, it is largest within the meaning of inclusion.

* Any subalgebra of a semi-simple algebra is semi-simple.

The demonstrations are in the article associated.

Lemma of Schur

Canonical decomposition

The decomposition of a semi-simple algebra in simple subalgebras is not single, to obtain a canonical decomposition, it is necessary to consider the relation of equivalence between the subalgebras simple data by the Isomorphisme S. Two algebras simple are in relation if and only if there exists an isomorphism of module between them. That is to say N_i the sum of the subalgebras of a given class. The following decomposition is canonical:

* N_i is the greatest subalgebra containing only isomorphous simple subalgebras with S _i and all its subalgebras are isomorphous with S _i. L is direct sum of N_i.

* With the preceding notations the N_i subalgebras are called isotypic factors L .

If an algebra L contains only subalgebra simple isomorphs two to two, then the algebra L is described as isotypic.

The Theorem of Artin-Wedderburn stated below, makes it possible to go further in the comprehension of the structure.

Analyzes endomorphisms

Simple ring

Let us determine the whole of the endomorphisms in a simple case, that of the ring regarded as a module on itself. Such a module is noted here _A has . A preliminary definition is necessary:

* the opposite ring of has is the ring noted here has ^op provided with the multiplication defined by:

\ forall has, B \ in \ mathbb has \ quad a^ {COp} .b^ {COp} =ba \;

One has the following property then:

* the whole of the endomorphisms of the module _A has is isomorphous with _A has ^op.

What amounts saying that the endomorphisms correspond to the opposite ring of has .

Indeed, if $\ phi$ is a endomorphism, it is entirely determined by the image of 1. Indeed, if has is an element of has , then its image is equal to has . φ (1), the following equality makes it possible to conclude:

\ forall \ varphi, \ psi \ in \ mathcal L_ {\ mathbb has} (\ mathbb A) \; \ forall has \ in \ mathbb has \ quad \ varphi \ circ \ psi (a)=a \ psi (1). \ varphi (1) \;

The structure of ring of the endomorphism is general to the algebras, in the case of a simple algebra, the lemma of Schur indicates that the endomorphism is either null, or invertible what shows the following property:

* the whole of the endomorphisms of a simple algebra is a body not necessarily commutative.

If the simple algebra is defined on a commutative body algebraically closed, the only automorphisms are the Homothétie S.

* the whole of the endormorphisms of a simple algebra L on a commutative body algebraically closed K is isomorphous in the center of the algebra and K .

This equality is also written: dim Hom_K^L ( L , L ) = 1 . This equality means that the algebra on the body K of the endomorphisms of L as an algebra is of dimension 1.

Simple algebra

Either L an element of L not no one, one uses the definition and of the following proposal:

* the whole of the elements has has such as has . L is equal zero is a maximum ideal bilataire, one calls it ideal canceler has because this ideal is independent of the element L not no one selected.

Either has an element of the ideal canceler of L . The homothety of report/ratio has has a core not no one, the lemma of Schur indicates whereas this homothety is the null application. The element has cancels any element of L , the cancelers of the various nonnull elements of L are thus all equal. This property also shows that the whole of the elements cancelling L is an ideal on the right. If B is an element of has , then B . has . L = B .0 = 0, the canceler is thus also an ideal on the left. Let us show finally that it is maximum, for that it is enough to notice that any element out of the canceler is invertible. It is still a direct consequence of the lemma of Schur. It is thus useless to consider the case of a simple algebra on a ring, it is enough to study that of a simple algebra on a body, in general left.

* the quotient of has by the ideal canceler is a body included in that of the opposite body of the endomorphisms of a simple algebra.

This quotient is made up of elements which are identified with invertible homotheties, it are thus many endomorphisms. A simple algebra on a body is the data of a simple ring L and of a subfield of the body of opposed algebra of the endomorphisms of L . For this reason, only the simple algebras on a body are studied.

Theorem of Artin-Wedderburn

Statement of the theorem

The theorem of Artin-Wedderburn is in the middle of the structure of the algebra, it expresses in the following way:

* a semi-simple ring such as any simple ideal is of size finished on its body of endomorphisms, is isomorphous with a product of algebras of endomorphisms of modules of dimension finished on bodies a priori distinct and left.

The analysis of the endomorphisms of a simple algebra shows that the theorem spreads immediately with the semi-simple algebras. Thus a simple algebra corresponds to a ring of endomorphisms of module on a left body, and an algebra corresponds to the same assistant structure of a subfield of the left body.

Reciprocally a generalization of a theorem of Burnside (cf paragraph Démonstration of Burnside of the article Théorème of Artin-Wedderburn) watch that the algebra of the endomorphisms of a module on a left body associated of the natural external multiplication on a subfield of the body defining the module is a simple algebra. Moreover, one end product of simple algebras is a semi-simple algebra. The reciprocal one of the theorem of Artin-Wedderburn is thus checked.

The unicity of the structure is ensured by the unicity of the decomposition of a semi-simple module in factors isotypic.

If has is a commutative body algebraically closed, then the preceding analysis shows that the bodies of the algebras of the endomorphisms are all equal to has . One profits then from the following proposal:

* If has is a commutative body algebraically closed and L a semi-simple algebra of finished size N , then the dimension of each factor isotypic is a perfect square D _i². Moreover the following equality is checked if H indicates the number of simple subalgebras included in L :

n=\sum_{i=1}^h d_i^2\;

Indeed, if D _i is the dimension of a simple module S _i of the I - ième isotypic component of L , then the I - ième algebra is that of the endomorphism of the subjacent vector space of S _i.

Center algebra

Let us study the center algebra if the algebra is defined on a commutative body algebraically closed. Are ( L _i) for I varying from 1 with H the family of the simple subalgebras and C an element of the center. Like the family of ( L _i) the form a direct sum equalizes with L , if one parallel to notes p _i the projector on L _i the direct sum of the other family members, one with the equality:

C = \ sum_ {ij \ in H} c_ {ij} \ quad with \ quad \ forall I, J \ in \ quad c_ {ij} = p_j \ circ C \ circ p_i \;

If I is different from J , then C _ij defines a morphism of L _i in L _j. The lemma of Schur indicates that such a morphism is always null. If I is equal to J , then C _ii corresponds to a morphism of simple algebra, it is thus a homothety of a report/ratio with value in K . In conclusion:

* the center of the algebra L is isomorphous with the ring summons direct of as many copies of K than there exist simple subalgebras.

This equality is written now if H indicates the numbers of subalgebras simple of the algebra L and C its center.

dim \, C = dim \; Hom_ {\ mathbb K} ^L (L \, \, L) = H \ quad and \ quad C \ simeq Hom_ {\ mathbb K} ^L (L \, \, L) \ simeq L^h

See too

External bonds

Théorie of algebras simple J.P. Tightens Seminar Henri Cartan
commutative Algèbre A. Chambert-Dormouse
Finite group pure representation for the mathematician Peter Webb
Théorie of the semi-simple algebras P. Cartier Séminaire Sophus Lie

References

NR. Bourbaki, commutative Algebra Chapter VIII and IX Masson 1983

Category: Commutative algebra Category: Theory of the representations Category: External structure Category: Linear algebra

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