Representation theory

Representation theory is a field of mathematics that examines abstract algebraic structures by expressing their elements as linear transformations of vector spaces. This approach makes abstract objects more tangible, often by representing their elements with matrices and their operations—like matrix addition and multiplication.

The algebraic structures that can be described this way include groups, associative algebras, and Lie algebras. Historically, the most prominent is the representation theory of groups, where group elements are shown as invertible matrices, and the group operation becomes matrix multiplication. This method is useful because it translates problems from abstract algebra into linear algebra, which is well understood. By representing more abstract objects in familiar linear algebra terms, properties become clearer and calculations simpler. For example, representing a group with an infinite-dimensional Hilbert space allows analytical techniques to be applied to group theory. Representation theory is also important in physics, as it describes how a physical system’s symmetry group influences the solutions to equations governing that system.

Representation theory appears throughout mathematics, with diverse applications. Beyond algebra, it generalizes Fourier analysis through harmonic analysis, connects to geometry via invariant theory and the Erlangen program, and impacts number theory through automorphic forms and the Langlands program. Multiple approaches exist for studying the same objects, drawing from algebraic geometry, module theory, analytic number theory, differential geometry, operator theory, algebraic combinatorics, and topology.

The success of representation theory has led to many generalizations, one of the most general being in category theory. The algebraic objects it applies to can be seen as particular kinds of categories, with representations as functors from the object category to the category of vector spaces. This suggests two natural extensions: first, the algebraic objects can be replaced by more general categories, and second, the target category of vector spaces can be replaced by other well-understood categories.

**Definitions and concepts**

Let \( V \) be a vector space over a field \( \mathbb{F} \). For instance, \( V \) could be \( \mathbb{R}^n \) or \( \mathbb{C}^n \), the standard n-dimensional space of column vectors over real or complex numbers. In this case, representation theory makes abstract algebra concrete using \( n \times n \) matrices of real or complex numbers. Three main types of algebraic objects can be handled this way: groups, associative algebras, and Lie algebras.

The set of all invertible \( n \times n \) matrices forms a group under matrix multiplication, and the representation theory of groups analyzes a group by describing its elements in terms of invertible matrices. Matrix addition and multiplication turn the set of all \( n \times n \) matrices into an associative algebra, leading to a representation theory of associative algebras. If matrix multiplication \( MN \) is replaced by the matrix commutator \( MN - NM \), the \( n \times n \) matrices become a Lie algebra, giving rise to a representation theory of Lie algebras.

This generalizes to any field \( \mathbb{F} \) and any vector space \( V \) over \( \mathbb{F} \), with linear maps replacing matrices and composition replacing matrix multiplication. There is a group \( \text{GL}(V, \mathbb{F}) \) of automorphisms of \( V \), an associative algebra \( \text{End}_{\mathbb{F}}(V) \) of all endomorphisms of \( V \), and a corresponding Lie algebra \( \mathfrak{gl}(V, \mathbb{F}) \).

**Definition**

**Action**

There are two ways to define a representation. The first uses the idea of an action, generalizing...