Category:Representation Theory

Representation theory is a branch of mathematics that deals with the study of symmetry and its mathematical representations. A representation of a mathematical object, such as a group, algebra, or Lie algebra, is a way of expressing it using a matrix or a linear operator. Representations can be used to study the properties of the original object, such as its symmetries, and to relate it to other objects. Representation theory has many applications in physics, mathematics, and computer science.

There are two main types of representations: finite-dimensional and infinite-dimensional. Finite-dimensional representations are used to study discrete symmetries, such as those of a crystal or a molecule. Infinite-dimensional representations are used to study continuous symmetries, such as those of a fluid or a field.

Representation theory of Lie groups and Lie algebras, is a powerful tool in theoretical physics, specially in particle physics and quantum field theory. Representations of Lie groups are used to describe the symmetries of physical systems and to classify elementary particles. Representations of Lie algebras are used to describe the symmetries of quantum systems and to classify states in quantum field theory.

The representation theory of finite groups is a vast area with many applications in many fields. It has been used to study the symmetries of mathematical objects such as knots and polynomials, and has applications in coding theory and cryptography.