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Concept# Semi-simplicity

Summary

In mathematics, semi-simplicity is a widespread concept in disciplines such as linear algebra, abstract algebra, representation theory, , and algebraic geometry. A semi-simple object is one that can be decomposed into a sum of simple objects, and simple objects are those that do not contain non-trivial proper sub-objects. The precise definitions of these words depends on the context.
For example, if G is a finite group, then a nontrivial finite-dimensional representation V over a field is said to be simple if the only subrepresentations it contains are either {0} or V (these are also called irreducible representations). Now Maschke's theorem says that any finite-dimensional representation of a finite group is a direct sum of simple representations (provided the characteristic of the base field does not divide the order of the group). So in the case of finite groups with this condition, every finite-dimensional representation is semi-simple. Especially in algebra and representation theory, "semi-simplicity" is also called complete reducibility. For example, Weyl's theorem on complete reducibility says a finite-dimensional representation of a semisimple compact Lie group is semisimple.
A square matrix (in other words a linear operator with V finite dimensional vector space) is said to be simple if its only invariant subspaces under T are {0} and V. If the field is algebraically closed (such as the complex numbers), then the only simple matrices are of size 1 by 1. A semi-simple matrix is one that is similar to a direct sum of simple matrices; if the field is algebraically closed, this is the same as being diagonalizable.
These notions of semi-simplicity can be unified using the language of semi-simple modules, and generalized to semi-simple .
If one considers all vector spaces (over a field, such as the real numbers), the simple vector spaces are those that contain no proper nontrivial subspaces. Therefore, the one-dimensional vector spaces are the simple ones.

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Related courses (7)

Related concepts (18)

Related lectures (44)

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Study the basics of representation theory of groups and associative algebras.

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Le but de ce cours est d'introduire et d'étudier les notions de base de l'algèbre abstraite.

Unipotent

In mathematics, a unipotent element r of a ring R is one such that r − 1 is a nilpotent element; in other words, (r − 1)n is zero for some n. In particular, a square matrix M is a unipotent matrix if and only if its characteristic polynomial P(t) is a power of t − 1. Thus all the eigenvalues of a unipotent matrix are 1. The term quasi-unipotent means that some power is unipotent, for example for a diagonalizable matrix with eigenvalues that are all roots of unity.

Semisimple representation

In mathematics, specifically in representation theory, a semisimple representation (also called a completely reducible representation) is a linear representation of a group or an algebra that is a direct sum of simple representations (also called irreducible representations). It is an example of the general mathematical notion of semisimplicity. Many representations that appear in applications of representation theory are semisimple or can be approximated by semisimple representations.

Semi-simplicity

In mathematics, semi-simplicity is a widespread concept in disciplines such as linear algebra, abstract algebra, representation theory, , and algebraic geometry. A semi-simple object is one that can be decomposed into a sum of simple objects, and simple objects are those that do not contain non-trivial proper sub-objects. The precise definitions of these words depends on the context. For example, if G is a finite group, then a nontrivial finite-dimensional representation V over a field is said to be simple if the only subrepresentations it contains are either {0} or V (these are also called irreducible representations).

Weyl character formulaMATH-680: Monstrous moonshine

Explores the proof of the Weyl character formula for finite-dimensional representations of semisimple Lie algebras.

Group Algebra: Maschke's TheoremMATH-334: Representation theory

Explores Wedderburn's theorem, group algebras, and Maschke's theorem in the context of finite dimensional simple algebras and their endomorphisms.

Isotypic DecompositionMATH-479: Linear algebraic groups

Covers the isotopic decomposition of modules into simple components and their properties.

This thesis deals with the study of G-forms and particulary the trace form of a G-Galois algebra. Let k be a field of characteristic not two. Let G be a finite group and L a G-Galois algebra over k. We define the trace form qL by qL(x, y) = TrL/k(xy) for all x, y in L. This is a bilinear symmetric form which is G-invariant. In other words, qL is a G-form. We know that L has a self-dual normal basis if and only if the trace form qL is G-isomorphic to the unit G-form q0. This is an important reason to classify the trace forms. This work contains two different parts. In the first part, we study G-forms in general, putting a ring structure on their Witt group. We then proved an analogue of Pfister's theorem - i.e. there is no zero divisor of odd dimension - when k[G] is semi-simple, k is big enough and G is abelian. Counter-examples are given when these conditions are not fulfilled. In the second part of this thesis, we study the trace form qL of a G-Galois algebra. E. Bayer-Fluckiger and H. W. Lenstra proved that if G is of odd order, then qL is always G-isomorphic to the unit form. If G is of even order, this is no longer the case. However, if the field k is of cohomological 2-dimension less than or equal to 1, then E. Bayer-Fluckiger and J.-P. Serre gave a necessary and sufficient condition - in terms of cohomological invariants - for the trace form qL to be isomorphic to the unit form. M. Monsurrò generalized this result to fields of virtual cohomological 2-dimension equal to 1. However, in higher cohomological dimensions, it becomes very difficult to classify the trace form itself. But it is possible to give general results if we consider multiples of the trace form or more generally the product of the trace form by a quadratic form. E. Bayer-Fluckiger formulated 2 conjectures about the possibility of finding a complete system of invariants for such a product when the quadratic form lies in a certain ideal of the Witt ring of k depending on the cohomological dimension of the field. In this work, we prove the first conjecture for all cohomological dimensions and the second one for a field of cohomological 2-dimension equal to 2. A more general conjecture is proved including the fields of virtual cohomological 2-dimension equal to 2. Finally, the second conjecture of E. Bayer-Fluckiger is proved in all cohomological dimensions, but only when either the characteristic of k is non zero or the group G is abelian or a 2-group, or k is big enough.