algebra simple — wikipedia

In mathematics, a algebra (associative unit) on a commutative body is said simple If its underlying ring is simple, that is to say if it does not admit a bilateral ideal other than {0} and itself, and if it is not reduced to 0. Whether A is a simple ring, so its center is a commutative body K , and considering A like an algebra on K , SO A is a simple algebra on K .

Thereafter, we designate by k a commutative body, and any algebra on K is supposed to be of finished dimension on K

An algebra on K is said central If it is not reduced to 0 and if its center is its subanary K .1.
Either A simple algebra on K . Then the center of A is an overly (commutative) WITH of K , A can be considered as an algebra on WITH , And A is a simple central algebra (in) on WITH . Thus, part of the study of simple algebras on a commutative body is reduced to the study of simple central algebras on a commutative body.

Here we are going to study the simple central algebras (or simple central algebras) on K .

Examples

• K is a simple central algebra on K .
• For any whole n ≥ 1, algebra M _n ( K ) square matrices is a simple central algebra.
• A K – Algebra (associative) with division ^{[ first ]} central (of finished dimension) is a simple central algebra. These algebras are the overflows D of K whose center is K and which are of finished dimension on K .
• Either D central division algebra on K . So, for any whole n ≥ 1, algebra M _n ( D ) Square matrices of order n on D is a simple central algebra on K . More intrinsically, for any vector space AND finished dimension on D , End algebra ( AND ) Endomorphisms of AND is a simple central algebra on K .

It is said that a simple central algebra on K East deployed ( split in English) if there is an integer n ≥ 1 such that A is isomorphic to algebra M _n ( K ) Square order matrices n .

Either A an algebra on K . It is equivalent to say that:

• A is a simple central algebra on K ;
• A is isomorphic to M _n ( D ), Or n ≥ 1 and D is an algebra with central division on K ;
• A is isomorphic to end algebra ( AND ) Endomorphisms of AND , Or AND is a non -zero finite dimension vector space on D , Or D is an algebra with central division on K .

Be D And D ‘Central division algebras on K , n And n ‘integers ≥ 1. so that algebra M _n ( D ) And M _{n ‘} ( D ‘) are isomorphic, it is necessary and it is enough that n = n ‘and that algebra D And D ‘are isomorphic.
Either AND And AND ‘Vector spaces of non -zero finished dimensions on D And D ‘ respectively. So that ends ends _D ( AND ) and ends _{D ‘} ( AND ‘) on K are isomorphic, it is necessary and it is enough that the algebra bodies D And D ‘ on K are isomorphic and that the dimensions of AND And AND ‘are equal.

The classification of simple central algebras on K is therefore reduced to the classification of central division on division on K .

Either A Simple central algebra and M A finished type module on A . Then K -Algèbre of the endomorphisms of M is a simple central algebra on K .

Examples of simple central algebras on certain commutative bodies [ modifier | Modifier and code ]

• On the body R Real numbers, simple central algebras are, with an isomorphism, the following: the algebras of the form M _n ( R ) and form M _n ( H ), Or H is the body of quaternions. More intrinsically, these are the algebras of the endomorphisms of real vector spaces and quaternionians of non -zero finished dimensions.
• And K is an algebraically closed body (for example if K is the body C with complex names) ou yes K is a finished body, the simple central algebras are, with a nearest isomorphisms, those of the form M _n ( K ).

Properties of simple power plants [ modifier | Modifier and code ]

Simple central algebras on K enjoy several remarkable properties.

• The opposite algebra of a simple central algebra on K is a simple central algebra.
• The tensorial product of two (or more generally of a finished family) of simple power plants on central K is a simple central algebra.
• And L is a commutative overflow of K , then algebra L ⊗ _K A on L deducted from A by extension of scalars of K To L is a simple central algebra.

Be A a simple central algebra on K And B simple algebra on K . Whatever homorphisms (unit) f And g of A In B , there is an invertible element b of B such as g ( x ) = bf ( x ) b ⁻¹ , for any element x From A ( f And g thereby are conjugated ).

In particular, any autumorphism of A is an inner self A , that is to say that it is in the form x ↦ axa ⁻¹ , Or a is an invertible element of A , and this autumorphism is then noted int _a . L’application a ↦ Int _a of the group A * Reversible elements of A In the AUT group _K ( A ) autumorphisms of K -algebra A is surjective, and its nucleus is the group K * non -zero scalars of K , and we thus obtain an isomorphism of groups of A * / K * on AUT _K ( A ).

Either D central division algebra on K And AND A finite dimension vector space n on K . Then the group of reversible END elements ( AND ) is the linear group GL ( AND ) of AND , and the application f ↦ Int _f of GL ( AND ) in Aut _K (End( AND )) is a core surjective homomorphism K , and we thus obtain an isomorphism of GL ( AND )/ K * on AUT _K (End( AND )). And n ≥ 2, then the group GL ( AND )/ K * is canonically isomorphic to the projective group of the projective space P ( AND ).

Either A a simple central algebra on K . Then the dimension of A on K is a square d ² , and we call degree of A the natural whole d .

Be A a simple central algebra on K And d the degree of A . There is a commutative overflow L of K such as the L -Pent central alternate L ⊗ _K A on L deducted from A by extension of scalars of K To L is deployed, that is to say isomorphic to M _d ( L ), and it is said that such an overflow L of K is a neutralizing body or one deployment of A .

Examples

• And A is deployed, then K is a neutralizing body of A .
• Any overly enclosed overorps of L (an algebraic fence of K for example) is a neutralizing body of A . For example, if K is the body R real numbers, C is a neutralizing body of A .

There is a neutralizing body L of A tel that the dimension of L is finished, and such that L (considered an extension of K ) is Galoisienne.

Either D central division algebra on K . So there is a maximum element L for the inclusion relationship of all the sub-bodys of D which are commutative. SO L is a neutralizing body of D , and more generally M _n ( D ). So, for any vector space AND finished dimension on D , L is a neutralizing body of end ( AND ).

With an element of a simple central algebra, we can associate scalars which generalize the trace, the determinant, and a polynomial which generalizes the characteristic polynomial, square matrices and endomorphisms of vector space on a commutative body.

Be A a simple central algebra on K , d the degree of A , L a neutralizing body of A And B = L ⊗ _K A the L -Alèbre simple central deducted from A by extension of scalars of K To L . For any element x of A and for any isomorphism of L -Algèbres h of B on M _d ( L ), the trace, the determinant and the polynomial characteristic of the matrix h (1 ⊗ x ) of M _d ( L ) only depend on A and of x (and not L or of h ), and they are called reduced trace , reduced standard And reduced characteristic polynomial of x In A (on K ), and we note them TRD _{A / K} ( x ), GDR _{A / K} ( x ) and PRD _{A / K} ( x ) respectively.

For example, if A = M _d ( K ) or A = End _K ( AND ), where E is a non -zero finished vector space on K , the reduced trace, the reduced standard and the characteristic polynomial reduced by an element of A are none other than its trace, its determinant and its characteristic polynomial.

In general :

• Function x ↦ Trd _{A / K} ( x ) of A In K is a linear form not identically zero in the vector space A .
• Whatever the elements a And b of A , we have nd _{A / K} ( ab ) = GDR _{A / K} ( a ) GDR _{A / K} ( b ).
• So that an element a of A be reversible in A , it is necessary and it is enough that NRD _{A / K} ( a ) be non -zero.
• Function x ↦ GDR _{A / K} ( x ) of the group A * Reversible elements of A In K * is a homomorphism of groups, not necessarily surjective. (He is overjective if A is deployed.)
• For any element a of A and for any element k of K , we have nd _{A / K} ( the ) = k ^d East Germany _{A / K} ( a ).
• If the body K is infinite, then the function x ↦ GDR _{A / K} ( x ) of A In K is a homogeneous degree polynomial function d .
• The degree of the polynomial characteristic of an element of a of A is equal to d , the reduced trace of a is the coefficient of X ^{n – first} and the reduced standard of a is the constant term, multiplied by (–1) ^d .

Trace and determining of an endomorphism of a quaternionnian vector space [ modifier | Modifier and code ]

Either AND A finite dimension vector space n on the body H Quaternions. Then the degree of A = End _H ( AND ) is 2 d . By restriction of scalars, we can consider AND Like a complex vector space AND ₀ , and then end _H ( AND ) is a real unitary sub-algebra of the simple END complex central algebra _C ( AND ₀ ). For all endomorphism f of AND , the reduced trace, the reduced standard and the characteristic polynomial reduced by the element f of A is none other than the trace, The norme ^{[Ref. necessary]} and the characteristic polynomial of the element f The End _C ( AND ₀ ).

Either f Endomorphism of AND . We call trace of f And we note TR f the reduced trace of f , divided by 2. The reduced standard of f is a real or zero real number ^{[Ref. necessary]} , and then we call determining of f an on note that f the square root of the reduced standard of f .

• N. Bourbaki, Algebra , chapter 8
• (in) Thomas W. Hungerford  (in) , Algebra , Springer-Verlag
• (in) Nathan Jacobson, Basic Algebra II , W. H. Freeman, New York, 1989
• (in) Max-Albert Kusus (of) , Alexander Mercury, Markus Rost (of) and Jean-Pierre Tignol, The Book of Involutions , AMS, 1998

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