Introduction to Representation Theory
Representation theory is a branch of mathematics that deals with the study of linear transformations between vector spaces. In particular, it focuses on the algebraic and geometric properties of these transformations, and how they can be used to study various mathematical objects such as groups, Lie algebras, and quantum mechanics.
The theory of representations is used in a wide range of fields, including physics, chemistry, computer science, and cryptography. It has applications in quantum mechanics, crystallography, and the study of symmetry in physical systems.
Representations of Groups
One of the primary applications of representation theory is the study of group theory. A group is a mathematical object that consists of a set of elements and a binary operation that combines two elements to produce another element in the set. The study of groups is used to describe and understand a wide range of physical phenomena, such as the symmetries of crystal structures and the rotational symmetry of molecules.
A representation of a group is a mapping of the group elements to linear transformations of a vector space. That is, for each group element, we associate a matrix that represents a linear transformation of the vector space. These matrices are required to satisfy certain properties to be considered a valid representation of the group.
One important property is that the identity element of the group must be represented by the identity matrix, and the inverse of a group element must be represented by the inverse of the corresponding matrix. The composition of two group elements must also be represented by the composition of the corresponding matrices.
Lie Algebras
Another application of representation theory is in the study of Lie algebras. A Lie algebra is a vector space equipped with a bilinear operation called the Lie bracket, which satisfies certain properties. Lie algebras are used to describe the infinitesimal symmetries of geometric objects such as curves, surfaces, and manifolds.
A representation of a Lie algebra is a mapping of the algebra elements to linear transformations of a vector space. The matrices corresponding to the Lie algebra elements must satisfy certain properties, including the requirement that the Lie bracket operation on the algebra is represented by the commutator of the corresponding matrices.
Quantum Mechanics
Representation theory is also used in the study of quantum mechanics. Quantum mechanics is a branch of physics that deals with the behavior of particles at the atomic and subatomic level. The theory of quantum mechanics is based on the concept of wave functions, which are represented by vectors in a Hilbert space.
In quantum mechanics, the symmetries of physical systems are described by unitary transformations, which are represented by matrices acting on the Hilbert space. These matrices must satisfy certain properties to be considered valid quantum mechanical transformations, including the requirement that they preserve the norm of the vectors in the Hilbert space.
Conclusion
In conclusion, representation theory is a powerful tool that is used in a wide range of mathematical and physical applications. It is used to study the symmetries of mathematical objects such as groups and Lie algebras, and to understand the behavior of physical systems in quantum mechanics. Representation theory is a fascinating subject that continues to be an active area of research in mathematics and physics.
Image of a group representation on a street art.