Lecture

Decomposition of Group Algebras

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In course

MATH-334: Representation theory

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

Description

This lecture covers examples of decomposition of group algebras into a direct sum of matrix algebras, the group algebra of a finite group as a left module over itself, and the concept of a simple infinite-dimensional algebra, specifically the Weyl algebra over a field of zero characteristic. The lecture also discusses theorems related to simple modules, such as Wedderburn's and Maschke's theorems, and provides examples of group algebras as left modules over themselves. Additionally, it explores the structure of the Weyl algebra, its basis, and its properties as a simple algebra that is not a matrix algebra. The lecture concludes with the study of tensor products in the context of group algebras.

Official source

Ontological neighbourhood

Mathematics

Algebra: Abstract algebra, Representation theory

Related concepts (32)

Simple module

In mathematics, specifically in ring theory, the simple modules over a ring R are the (left or right) modules over R that are non-zero and have no non-zero proper submodules. Equivalently, a module M is simple if and only if every cyclic submodule generated by a non-zero element of M equals M. Simple modules form building blocks for the modules of finite length, and they are analogous to the simple groups in group theory. In this article, all modules will be assumed to be right unital modules over a ring R.

Module (mathematics)

In mathematics, a module is a generalization of the notion of vector space in which the field of scalars is replaced by a ring. The concept of module generalizes also the notion of abelian group, since the abelian groups are exactly the modules over the ring of integers. Like a vector space, a module is an additive abelian group, and scalar multiplication is distributive over the operation of addition between elements of the ring or module and is compatible with the ring multiplication.

Semisimple module

In mathematics, especially in the area of abstract algebra known as module theory, a semisimple module or completely reducible module is a type of module that can be understood easily from its parts. A ring that is a semisimple module over itself is known as an Artinian semisimple ring. Some important rings, such as group rings of finite groups over fields of characteristic zero, are semisimple rings. An Artinian ring is initially understood via its largest semisimple quotient.

Algebraic group

In mathematics, an algebraic group is an algebraic variety endowed with a group structure that is compatible with its structure as an algebraic variety. Thus the study of algebraic groups belongs both to algebraic geometry and group theory. Many groups of geometric transformations are algebraic groups; for example, orthogonal groups, general linear groups, projective groups, Euclidean groups, etc. Many matrix groups are also algebraic. Other algebraic groups occur naturally in algebraic geometry, such as elliptic curves and Jacobian varieties.

Projective module

In mathematics, particularly in algebra, the class of projective modules enlarges the class of free modules (that is, modules with basis vectors) over a ring, by keeping some of the main properties of free modules. Various equivalent characterizations of these modules appear below. Every free module is a projective module, but the converse fails to hold over some rings, such as Dedekind rings that are not principal ideal domains.

Related lectures (64)

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Density Theorem: Endomorphisms of Simple Modules

Covers the density theorem in representation theory, focusing on endomorphisms of simple finite-dimensional modules.

Ideals and Representations

Covers ideals, representations, modules, and maximal ideals in associative algebras.

Structure of Algebras

Covers the structure of finite dimensional algebras and the characterization of semisimple algebras.

Tensor Products of Modules