Semisimple module

As a semisimple is called in mathematics certain structures, which are composed of comparatively easy to understand way of " building blocks ".

The term is used in the mathematical field of algebra in different contexts. He has particular importance in the theory of modules and rings. The " building blocks " are the simple modules here. Then the semi simple modules a certain extent make the nächstkompliziertere stage, namely, those which are assembled by means of direct sum of the simple modules. About semisimple modules ( and rings ) are many sets known, they are mathematically so, as the name suggests, still quite "simple" objects.

One of the most important applications is in the representation theory of groups and is based on the theorem of Maschke.

  • 2.1 Definition
  • 2.2 Features
  • 2.3 set of Artin - Wedderburn
  • 3.1 Linear maps
  • 3.2 matrices
  • 3.3 associated with semisimple algebras

Semi- Simple module

Definition

( In the following, familiarity of the reader is provided with the concept of the module. )

Be a module over a ring ( with unity).

The module is called semi- simple or completely reducible if one of the following equivalent conditions is satisfied:

Properties

  • Submodules, quotient modules and direct sums of semi- simple modules are semisimple.
  • A module is semisimple and finitely generated if and only if it is artinian and its Jacobson radical.

Examples

  • The finitely generated semisimple -modules are precisely the direct sums of modules of the form of square-free numbers.
  • Is a body, a module is nothing more than a vector space. These are always semisimple.

Semi- Simple rings

Each ring acts on itself by multiplication from the left and thus becomes a left module over itself The submodules are then precisely the left ideals. The irreducible submodules are precisely the non-trivial minimal left ideals. Of course you can make analogous to a right module over itself.

Definition

A ring is called semisimple if it is semisimple as a module over itself. One can show that this is not dependent on whether you look at as a left or right module.

Note: A ring is called simple if it has no nontrivial two-sided ideals (and not when he's just about yourself as a module). Not every simple ring is semisimple. This terminology is confusing, but has prevailed.

Properties

  • A unitary ring is semisimple if and only if it is artinian and its Jacobson radical. ( This is a special case of the above property for semisimple modules, as is generated as a module over itself of the. )
  • In particular, the factor ring is semisimple artinian for a ring.
  • Is semisimple, then every module is semisimple. This follows from the above properties of semi simple modules and from the fact that each module is a quotient of a free module ( that is, a direct sum of copies ) is.
  • About semisimple rings all modules are projective.

Set of Artin - Wedderburn

Every semisimple ring is isomorphic to a (finite) direct product of matrix rings over skew fields. Here the complete matrix ring is meant not a subring.

Semi- Simple matrices

Linear maps

Be a vector space. A linear mapping is called semisimple if there is a basis of, in the represented by a diagonal matrix.

The mapping is called - semisimple or hyperbolic if there is a basis of, in the represented by a diagonal matrix. The mapping is called - semisimple or elliptical if it is semisimple and all eigenvalues ​​of magnitude 1. Each linear mapping can be uniquely as a product of a - semisimple, disassemble a unipotent and semisimple a - figure, see Iwasawa decomposition.

Matrices

A matrix is called semisimple if the associated linear map is semisimple.

The following conditions are equivalent:

  • Is semisimple,
  • Is diagonalizable,
  • The minimal polynomial of has no multiple factors.

Associated with semisimple algebras

A matrix is ​​then exactly semisimple if a semisimple algebra.

Example: Application in representation theory

Let be a finite group and a body. Be the group algebra (it is the vector space with basis and the multiplication is induced by the group structure). The representations of in - vector spaces correspond exactly to the moduli. Sub-images correspond to sub- modules and irreducible representations correspond to simple modules.

Let now such that the characteristic does not divide by (eg). Then the set of states Maschke that the group algebra and let everyone module is semisimple.

Group representation Maschke's theorem Artin group Square-free integer Quotient ring Projective object Minimal polynomial (linear algebra) Monoidring Characteristic (algebra)
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