Group of Brauer

In Mathematical, the group of Brauer constitutes the space classifying of the simple central algebras on a commutative body K given, for a certain relation of equivalence. One provides this space with a structure of abelian Groupe by identifying it with a space of Cohomologie galoisienne.

Construction of the group of Brauer

A simple central algebra on a commutative body K , is a associative algebra of finished size has , which does not admit any noncommonplace ideal bilatère (simplicity), and whose center is K (centrality). For example, the body of the complex numbers forms a simple central algebra on itself, but not on the body of the real numbers, the property of centrality being at fault. On the other hand, the algebra of the quaternions of Hamilton is a simple central algebra on the body of the real numbers.

Being given two simple central algebras has and B , one defines the tensorial product $has\; \backslash \; otimes\; B$ starting from the tensorial product of spaces vectorielspris like vector spaces by adding the property of bilinearity: $(has\; \backslash \; otimes\; b)\; \backslash \; cdot\; (C\; \backslash \; otimes\; d)\; =\; ac\; \backslash \; otimes\; bd.$ a tensorial product of two simple central algebras is a simple central algebra.

The first important characterization of the simple central algebras is that in fact exactly the algebras has of finished size become isomorphous with an algebra of matrices M_n (K) by extension of the scalars to a finished extension K of the body K ; i.e. by considering the tensorial product $A\; \backslash \; otimes\_k\; K\; \backslash \; simeq\; M\_n\; (K)$. In addition, the theorem of Wedderburn ensures that any simple algebra is isomorphous with an algebra of matrices with coefficients in a body (noncommutative) D container K , the body D being single except for isomorphism. The following relation then is introduced: two simple algebras central has and A' is equivalent if and only if the same body D can be selected for both in what precedes. Another equivalent definition consists in requiring that there exist entireties m and N such as one has an isomorphism of algebras $A\; \backslash \; otimes\_k\; M\_m\; (K)\; \backslash \; simeq\; A\text{'}\; \backslash \; otimes\_k\; M\_n\; (K)$.

The classes of equivalence for this relation then form an abelian group for the tensorial product called group of Brauer . The opposite of a class of equivalence whose representative is has is the class of equivalence of the opposite Algèbre A^op (defined by changing the operation of multiplication . by the relation * defined by: a*b=b.a); this shows by fact that morphisme which with $a\; \backslash \; otimes\; b$ associates the K - endomorphism on has which with X associates axb defines an isomorphism between the tensorial product $A\; \backslash \; otimes\_k\; A\text{'}$ and a space of K - endomorphisms, i.e. a space of matrices with coefficients in K , therefore commonplace in the group of Brauer.

Examples

The group of Brauer of a Body algebraically closed (all the simple central algebras are isomorphous with an algebra of matrices) or of a Corps finished (the bodies containing a finished body, and of size finished on this one are thus finished commutative by another Théorème of Wedderburn) is the commonplace Groupe.

The group of Brauer $Br\; (\backslash \; mathbb\; \{R\})$ of the body of the real numbers $\backslash \; mathbb\; \{R\}$ is a cyclic Groupe of order two: there exist only two types of body containing that of the real numbers and exchange on this one, namely $\backslash \; mathbb\; \{R\}$ itself and the algebra of the Quaternion S $\backslash \; mathbb\; \{H\}\; \backslash ,$. The product in the group of Brauer is based on the tensorial Produit: the statement that $\backslash \; mathbb\; \{H\}\; \backslash ,$ is of order two in the group of Brauer is equivalent to the existence of an isomorphism of $\backslash \; mathbb\; \{R\}$-algèbres

$\backslash \; mathbb\; \{H\}\; \backslash \; otimes\; \backslash \; mathbb\; \{H\}\; \backslash \; cong\; M\; (4,\; \backslash \; mathbb\; \{R\})$

algebras with 16 dimensions.

Generalization

The groups of Brauer of the local bodies can be calculated; they all are canonically isomorphous with $\backslash \; mathbb\; \{Q\}/\backslash \; mathbb\; \{Z\}\; \backslash ,$, for the p-adic Corps of numbers. The results are then applied to the total bodies, it is the cohomologic approach of the Théorie of the bodies of classes. More precisely, the group of Brauer Br ( K ) of a total body K is given thereafter exact

$0\; \backslash \; rightarrow\; Br\; (K)\; \backslash \; rightarrow\; \backslash \; oplus\_p\; Br\; (K\_p)\; \backslash \; rightarrow\; \backslash \; mathbb\; \{Q\}/\backslash \; mathbb\; \{Z\}\; \backslash \; rightarrow\; 0$

where the sum of the medium relates to all completions (archimédiennes and non-archimédiennes) of K . The group $\backslash \; mathbb\; \{Q\}/\backslash \; mathbb\; \{Z\}\; \backslash ,$ on the line is in fact the " group of Brauer" Formation of classes of the classes of idèles associated with K .

In the general theory, the group of Brauer is expressed by groups of cohomology:

$Br\; (K)\; \backslash \; cong\; H^2\; (Gall\; (k^s/k),\; \{k^s\}\; ^*),$

where K ^S is the separable Clôture body K .

A generalization in algebraic Geometry, due to Grothendieck, constitutes the theory of the algebras of Azumaya.

See too

• simple central Algebra

• Formation of classes
• Variety of Severi-Brauer

External bonds

• PlanetMath page

• MathWorld page

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