Representation theory
Introduction
Representation theory is a branch of mathematics that studies abstract algebraic structures by representing their elements as linear transformations of vector spaces. This field of study is used in various areas of mathematics, including algebra, number theory, and physics.
Historical Background
The concept of representation was first introduced by the German mathematician Frobenius in 1896, in the context of group theory. His work laid the foundation for the development of representation theory, which has since been expanded and applied in various fields of mathematics and physics.
Basic Concepts
Representation theory is based on several fundamental concepts, including groups, rings, and modules.
Groups
A group is a set of elements together with an operation that combines any two of its elements to form a third element. In the context of representation theory, groups are often represented as matrices, which can be thought of as arrays of numbers that can be multiplied together.
Rings
A ring is an algebraic structure that consists of a set equipped with two binary operations, usually referred to as addition and multiplication. Rings are fundamental objects of study in representation theory, as they provide a framework for understanding the structure of other algebraic systems.
Modules
A module is a generalization of the notion of vector space. In representation theory, modules are used to study the structure of rings and other algebraic systems.
Applications
Representation theory has wide-ranging applications in various fields of mathematics and physics.
Algebra
In algebra, representation theory is used to study the structure of algebraic systems such as groups, rings, and fields. By representing these systems as matrices or linear transformations, it is possible to gain insights into their structure and properties.
Number Theory
In number theory, representation theory is used to study the distribution of prime numbers and the solutions of Diophantine equations. The theory of automorphic forms, for instance, relies heavily on the techniques of representation theory.
Physics
In physics, representation theory is used in the study of quantum mechanics and quantum field theory. The theory of quantum mechanics, for instance, is built on the representation theory of the Heisenberg and Poincaré groups.