Semi-simple module

In mathematics , semi-simple is the term used to describe certain structures that are composed of “basic building blocks” in a comparatively easy-to-understand way.

The term is used in different contexts in the mathematical field of algebra . It is of particular importance in the theory of modules and rings . The "basic building blocks" here are the simple modules . The semi-simple modules then to a certain extent form the next more complicated level, namely those that are composed of simple modules by means of a direct sum . Many theorems are known about semi-simple modules (and rings), so mathematically speaking, as the name suggests, they are still quite “simple” objects.

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

Semi-simple module

definition

(In the following it is assumed that the reader is familiar with the term module .)

Be a module over a ring (with one) . ${\displaystyle M}$ ${\displaystyle R}$

The module is called semi-simple or fully reducible if one of the following equivalent conditions is met: ${\displaystyle M}$

1. ${\displaystyle M}$can be written as the direct sum of simple modules .
2. ${\displaystyle M}$ can be written as the sum of simple modules.
3. Existence of complements: For every sub-module of there exists a sub-module of such that . ${\displaystyle N}$ ${\displaystyle M}$ ${\displaystyle P}$ ${\displaystyle M}$ ${\displaystyle M\simeq N\oplus P}$

properties

• Sub- modules , quotient modules, and direct sums of semi-simple modules are semi-simple.
• A module is semi-simple and finitely generated if and only if it is Artinian and its Jacobson radical is. ${\displaystyle M}$ ${\displaystyle Rad(M)=0}$

Examples

• The finitely generated semisimple -modules are exactly the direct sums of modules of the form for square-free numbers . ${\displaystyle \mathbb {Z} }$ ${\displaystyle \mathbb {Z} /n\mathbb {Z} }$ ${\displaystyle n}$
• If there is a body , then a module is nothing more than a vector space . These are always semi-simple. ${\displaystyle R}$ ${\displaystyle R}$

Semi-simple rings

Each ring acts on itself through multiplication from the left and thus becomes a link module over itself. The sub-modules are then exactly the left ideals . The irreducible sub-modules are precisely the nontrivial minimal left ideals. Of course, you can do about yourself in the same way as a legal module. If the ring is commutative, the two constructions agree with each other and result in the same structure. ${\displaystyle R}$ ${\displaystyle R}$

definition

A ring is called semi-simple if it is semi-simple as a module about itself. One can show that this does not depend on whether one considers a left or right module. ${\displaystyle R}$

Comment: A ring is called simple if it has no nontrivial mutual ideals (and not if it is simple as a module about itself). Not every simple ring is semi-simple. This terminology is confusing, but it has caught on.

properties

• A unitary ring is semi-simple if and only if it is Artinian and its Jacobson radical is. (This is a special case of the above property for semi-simple modules, because it is created as a module over itself by the .) ${\displaystyle R}$ ${\displaystyle Jac(R)=0}$ ${\displaystyle R}$ ${\displaystyle 1}$
• In particular, for an Artin's ring, the factor ring is semi- simple. ${\displaystyle R}$ ${\displaystyle R/Jac(R)}$
• If semi-simple, every module is semi-simple. This follows from the above properties of semi-simple modules and from the fact that each module is a quotient of a free module (i.e. a direct sum of copies of ). ${\displaystyle R}$ ${\displaystyle R}$ ${\displaystyle R}$
• All modules are projective over semi-simple rings .

Artin-Wedderburn's theorem

Every semi-simple ring is isomorphic to a (finite) direct product of matrix rings over skew fields . The whole die ring is meant here, not a sub-ring.

Semi-simple matrices

Linear maps

Let be a vector space. A linear mapping is half simply , if there is a basis of there, in by a diagonal matrix is illustrated. ${\displaystyle V}$ ${\displaystyle \mathbb {C} }$ ${\displaystyle f\colon V\rightarrow V}$ ${\displaystyle \mathbb {C} }$ ${\displaystyle V}$ ${\displaystyle f}$

The figure is -halbeinfach or hyperbolic if there is a basis of there, in the by a diagonal matrix is presented. The mapping is called -simple or elliptic if it is semisimple and all eigenvalues ​​have magnitude 1. Every linear mapping can be uniquely decomposed as the product of a -simple, unipotent and -simple mapping, see Iwasawa decomposition . ${\displaystyle \mathbb {R} }$ ${\displaystyle \mathbb {R} }$ ${\displaystyle V}$ ${\displaystyle f}$ ${\displaystyle S^{1}}$ ${\displaystyle S^{1}}$ ${\displaystyle \mathbb {R} }$

Matrices

A matrix is called semi-simple if the associated linear map is semi-simple. ${\displaystyle A\in Mat(n,\mathbb {C} )}$ ${\displaystyle f\colon \mathbb {C} ^{n}\rightarrow \mathbb {C} ^{n}}$

The following conditions are equivalent:

• ${\displaystyle A\in Mat(n,\mathbb {C} )}$ is semi-easy,
• ${\displaystyle A\in Mat(n,\mathbb {C} )}$is diagonalizable ,
• the minimal polynomial of has no multiple factors. ${\displaystyle A\in Mat(n,\mathbb {C} )}$

Relation to semi-simple algebras

A matrix is semi-simple if and only if there is a semi-simple algebra. ${\displaystyle A\in Mat(n,\mathbb {C} )}$ ${\displaystyle \mathbb {C} \left[A\right]}$

Example: application in representation theory

Be a finite group and a body . Let be the group algebra (this is the -vector space with base and the multiplication induced by the group structure). The representations of in vector spaces correspond exactly to the modules. Sub-representations correspond to sub-modules and irreducible representations correspond to simple modules. ${\displaystyle G}$ ${\displaystyle K}$ ${\displaystyle K[G]}$ ${\displaystyle K}$ ${\displaystyle G}$ ${\displaystyle G}$ ${\displaystyle K}$ ${\displaystyle K[G]}$

Now be such that the characteristic of does not divide (e.g. ). Then Maschke's theorem says that group algebra and thus every module is semi-simple. ${\displaystyle K}$ ${\displaystyle K}$ ${\displaystyle |G|}$ ${\displaystyle K=\mathbb {C} }$ ${\displaystyle K[G]}$ ${\displaystyle K[G]}$

See also

• Semi-simple Lie algebra
• Semi-simple lie group

literature

• Serge Lang, algebra
• Nathan Jacobson, Basic Algebra II