Representation theory is a branch of mathematics that studies abstract algebraic structures by representing their elements as linear transformations of vector spaces.Classic texts on representation theory include and . Other excellent sources are and . In essence, a representation makes an abstract algebraic object more concrete by describing its elements by matrices and their algebraic operations (for example, matrix addition, matrix multiplication).
The algebraic objects amenable to such a description include groups, associative algebras and Lie algebras. The most prominent of these (and historically the first) is the representation theory of groups, in which elements of a group are represented by invertible matrices such that the group operation is matrix multiplication.For the history of the representation theory of finite groups, see . For algebraic and Lie groups, see .
Representation theory is a useful method because it reduces problems in abstract algebra to problems in linear algebra, a subject that is well understood. There are many textbooks on vector spaces and linear algebra. For an advanced treatment, see . Representations of more abstract objects in terms of familiar linear algebra can elucidate properties and simplify calculations within more abstract theories. For instance, representing a group by an infinite-dimensional Hilbert space allows methods of analysis to be applied to the theory of groups. Furthermore, representation theory is important in physics because it can describe how the symmetry group of a physical system affects the solutions of equations describing that system.
Representation theory is pervasive across fields of mathematics. The applications of representation theory are diverse. In addition to its impact on algebra, representation theory generalizes Fourier analysis via harmonic analysis, is connected to geometry via invariant theory and the Erlangen program, has an impact in number theory via automorphic forms and the Langlands program.
There are many approaches to representation theory: the same objects can be studied using methods from algebraic geometry, module theory, analytic number theory, differential geometry, operator theory, algebraic combinatorics and topology.See the previous footnotes.
The success of representation theory has led to numerous generalizations. One of the most general is in category theory. The algebraic objects to which representation theory applies can be viewed as particular kinds of categories, and the representations as functors from the object category to the category of vector spaces. This description points to two natural generalizations:
First, the algebraic objects can be replaced by more general categories. Second, the target category of vector spaces can be replaced by other well-understood categories.
Definitions and concepts
Let be a vector space over a field . For instance, suppose is or , the standard -dimensional space of vectors over the real or complex numbers, respectively. In this case, the idea of representation theory is to do abstract algebra concretely by using matrices of real or complex numbers.
There are three main sorts of algebraic objects for which this can be done: groups, associative algebras and Lie algebras.
The set of all invertible matrices is a group under matrix multiplication, and the representation theory of groups analyzes a group by describing ("representing") its elements in terms of invertible matrices. Matrix addition and multiplication make the set of all matrices into an associative algebra, and hence there is a corresponding representation theory of associative algebras. If we replace matrix multiplication by the matrix commutator , then the matrices become instead a Lie algebra, leading to a representation theory of Lie algebras.
This generalizes to any field and any vector space over , with linear maps replacing matrices and composition replacing matrix multiplication: there is a group of automorphisms of , an associative algebra of all endomorphisms of , and a corresponding Lie algebra .
Definition
Action There are two ways to define a representation.This material can be found in standard textbooks, such as , , , , , , and . The first uses the idea of an action, generalizing the way that matrices act on column vectors by matrix multiplication.
A representation of a group or (associative or Lie) algebra on a vector space is a map
with two properties:
For any in (or in ), the map is linear (over ). If we introduce the notation for , then for any , in and in :
where is the identity element of and is the group product in . The definition for associative algebras is analogous, except that associative algebras do not always have an identity element, in which case equation (2.1) is omitted. Equation (2.2) is an abstract expression of the associativity of matrix multiplication. This doesn't hold for the matrix commutator and also there is no identity element for the commutator. Thus, for Lie algebras, the only requirement is that for any , in and in ,
where is the Lie bracket, which generalizes the matrix commutator .
Mapping The second way to define a representation focuses on the map sending in to a linear map , which satisfies
and similarly in the other cases. This approach is both more concise and more abstract. From this point of view: a representation of a group on a vector space is a group homomorphism ; a representation of an associative algebra on a vector space is an algebra homomorphism ; a representation of a Lie algebra on a vector space is a Lie algebra homomorphism .
Terminology The vector space is called the representation space of and its dimension (if finite) is called the dimension (sometimes degree) of the representation. It is also common practice to refer to itself as the representation when the homomorphism is clear from context; otherwise the notation can be used to denote a representation.
When is of finite dimension , one can choose a basis for to identify with , and hence recover a matrix representation with entries in the field .
An effective or faithful representation is a representation for which the homomorphism is injective.
Equivariant maps and isomorphisms
If and are vector spaces over , equipped with representations and of a group , then an equivariant map from to is a linear map such that
for all in and in . In terms of and , this means
for all in ; that is, the following diagram commutes:
Equivariant maps for representations of an associative or Lie algebra are defined similarly. If is invertible, then it is said to be an isomorphism, in which case and (or, more precisely, and ) are isomorphic representations, also phrased as equivalent representations. An equivariant map is often called an intertwining map of representations. Also, in the case of a group , it is on occasion called a -map or -linear map.
Isomorphic representations are, for practical purposes, "the same"; they provide the same information about the group or algebra being represented. Representation theory therefore seeks to classify representations up to isomorphism.
Subrepresentations, quotients, and irreducible representations
If is a representation of (say) a group , and is a linear subspace of that is preserved by the action of in the sense that for all and , (Serre calls these stable under ), then is called a subrepresentation: by defining where is the restriction of to , is a representation of and the inclusion of is an equivariant map. The quotient space can also be made into a representation of . If has exactly two subrepresentations, namely the trivial subspace and itself, then the representation is said to be irreducible; if has a proper nontrivial subrepresentation, the representation is said to be reducible.The representation {0} of dimension zero is considered to be neither reducible nor irreducible, just like the number 1 is considered to be neither composite nor prime.
The definition of an irreducible representation implies Schur's lemma: an equivariant map between irreducible representations is either the zero map or an isomorphism, since its kernel and image are subrepresentations. In particular, when , this shows that the equivariant endomorphisms of form an associative division algebra over the underlying field . If is algebraically closed, the only equivariant endomorphisms of an irreducible representation are the scalar multiples of the identity.
Irreducible representations are the building blocks of representation theory for many groups: if a representation is not irreducible then it is built from a subrepresentation and a quotient that are both "simpler" in some sense; for instance, if is finite-dimensional, then both the subrepresentation and the quotient have smaller dimension. There are counterexamples where a representation has a subrepresentation, but only has one non-trivial irreducible component. For example, the additive group has a two-dimensional representation
This group has the vector fixed by this homomorphism, but the complement subspace maps to
giving only one irreducible subrepresentation. This is true for all unipotent groups.
Direct sums and indecomposable representations
If and are representations of (say) a group , then the direct sum of and is a representation, in a canonical way, via the equation
The direct sum of two representations carries no more information about the group than the two representations do individually. If a representation is the direct sum of two proper nontrivial subrepresentations, it is said to be decomposable. Otherwise, it is said to be indecomposable.
Complete reducibility In favorable circumstances, every finite-dimensional representation is a direct sum of irreducible representations: such representations are said to be semisimple. In this case, it suffices to understand only the irreducible representations. Examples where this "complete reducibility" phenomenon occurs (at least over fields of characteristic zero) include finite groups (see Maschke's theorem), compact groups, and semisimple Lie algebras.
In cases where complete reducibility does not hold, one must understand how indecomposable representations can be built from irreducible representations by using extensions of quotients by subrepresentations.
Tensor products of representations
Suppose and are representations of a group . Then we can form a representation of acting on the tensor product vector space as follows: . If and are representations of a Lie algebra, then the correct formula to use is .
This product can be recognized as the coproduct on a coalgebra. In general, the tensor product of irreducible representations is not irreducible; the process of decomposing a tensor product as a direct sum of irreducible representations is known as Clebsch–Gordan theory.
In the case of the representation theory of the group SU(2), or, equivalently, of its complexified Lie algebra , the decomposition is easy to work out. The irreducible representations are labeled by a parameter that is a non-negative integer or half integer; the representation then has dimension . Suppose we take the tensor product of the representation of two representations, with labels and where we assume . Then the tensor product decomposes as a direct sum of one copy of each representation with label , where ranges from to in increments of 1. If, for example, , then the values of that occur are 0, 1, and 2. Thus, the tensor product representation of dimension decomposes as a direct sum of a 1-dimensional representation a 3-dimensional representation and a 5-dimensional representation .
Branches and topics