In
mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
, a unitary representation of a
group
A group is a number of persons or things that are located, gathered, or classed together.
Groups of people
* Cultural group, a group whose members share the same cultural identity
* Ethnic group, a group whose members share the same ethnic ide ...
''G'' is a
linear representation
Representation theory is a branch of mathematics that studies abstract algebraic structures by ''representing'' their elements as linear transformations of vector spaces, and studies modules over these abstract algebraic structures. In essenc ...
π of ''G'' on a complex
Hilbert space
In mathematics, Hilbert spaces (named after David Hilbert) allow generalizing the methods of linear algebra and calculus from (finite-dimensional) Euclidean vector spaces to spaces that may be infinite-dimensional. Hilbert spaces arise natural ...
''V'' such that π(''g'') is a
unitary operator
In functional analysis, a unitary operator is a surjective bounded operator on a Hilbert space that preserves the inner product. Unitary operators are usually taken as operating ''on'' a Hilbert space, but the same notion serves to define the co ...
for every ''g'' ∈ ''G''. The general theory is well-developed in case ''G'' is a
locally compact In topology and related branches of mathematics, a topological space is called locally compact if, roughly speaking, each small portion of the space looks like a small portion of a compact space. More precisely, it is a topological space in which ev ...
(
Hausdorff)
topological group
In mathematics, topological groups are logically the combination of groups and topological spaces, i.e. they are groups and topological spaces at the same time, such that the continuity condition for the group operations connects these two st ...
and the representations are
strongly continuous.
The theory has been widely applied in
quantum mechanics
Quantum mechanics is a fundamental theory in physics that provides a description of the physical properties of nature at the scale of atoms and subatomic particles. It is the foundation of all quantum physics including quantum chemistry, ...
since the 1920s, particularly influenced by
Hermann Weyl
Hermann Klaus Hugo Weyl, (; 9 November 1885 – 8 December 1955) was a German mathematician, theoretical physicist and philosopher. Although much of his working life was spent in Zürich, Switzerland, and then Princeton, New Jersey, he is asso ...
's 1928 book ''Gruppentheorie und Quantenmechanik''. One of the pioneers in constructing a general theory of unitary representations, for any group ''G'' rather than just for particular groups useful in applications, was
George Mackey
George Whitelaw Mackey (February 1, 1916 – March 15, 2006) was an American mathematician known for his contributions to quantum logic, representation theory, and noncommutative geometry.
Career
Mackey earned his bachelor of arts at Rice Un ...
.
Context in harmonic analysis
The theory of unitary representations of topological groups is closely connected with
harmonic analysis
Harmonic analysis is a branch of mathematics concerned with the representation of functions or signals as the superposition of basic waves, and the study of and generalization of the notions of Fourier series and Fourier transforms (i.e. an ex ...
. In the case of an abelian group ''G'', a fairly complete picture of the representation theory of ''G'' is given by
Pontryagin duality
In mathematics, Pontryagin duality is a duality (mathematics), duality between locally compact abelian groups that allows generalizing Fourier transform to all such groups, which include the circle group (the multiplicative group of complex numb ...
. In general, the unitary equivalence classes (see
below
Below may refer to:
*Earth
*Ground (disambiguation)
*Soil
*Floor
*Bottom (disambiguation)
*Less than
*Temperatures below freezing
*Hell or underworld
People with the surname
*Ernst von Below (1863–1955), German World War I general
*Fred Below ( ...
) of
irreducible unitary representations of ''G'' make up its unitary dual. This set can be identified with the
spectrum of the C*-algebra associated to ''G'' by the
group C*-algebra construction. This is a
topological space
In mathematics, a topological space is, roughly speaking, a geometrical space in which closeness is defined but cannot necessarily be measured by a numeric distance. More specifically, a topological space is a set whose elements are called poin ...
.
The general form of the
Plancherel theorem
In mathematics, the Plancherel theorem (sometimes called the Parseval–Plancherel identity) is a result in harmonic analysis, proven by Michel Plancherel in 1910. It states that the integral of a function's squared modulus is equal to the integ ...
tries to describe the regular representation of ''G'' on ''L''
2(''G'') by means of a
measure on the unitary dual. For ''G'' abelian this is given by the Pontryagin duality theory. For ''G''
compact
Compact as used in politics may refer broadly to a pact or treaty; in more specific cases it may refer to:
* Interstate compact
* Blood compact, an ancient ritual of the Philippines
* Compact government, a type of colonial rule utilized in Britis ...
, this is done by the
Peter–Weyl theorem
In mathematics, the Peter–Weyl theorem is a basic result in the theory of harmonic analysis, applying to topological groups that are compact, but are not necessarily abelian. It was initially proved by Hermann Weyl, with his student Fritz Peter, ...
; in that case the unitary dual is a
discrete space
In topology, a discrete space is a particularly simple example of a topological space or similar structure, one in which the points form a , meaning they are ''isolated'' from each other in a certain sense. The discrete topology is the finest top ...
, and the measure attaches an atom to each point of mass equal to its degree.
Formal definitions
Let ''G'' be a topological group. A strongly continuous unitary representation of ''G'' on a Hilbert space ''H'' is a group homomorphism from ''G'' into the unitary group of ''H'',
:
such that ''g'' → π(''g'') ξ is a norm continuous function for every ξ ∈ ''H''.
Note that if G is a
Lie group
In mathematics, a Lie group (pronounced ) is a group that is also a differentiable manifold. A manifold is a space that locally resembles Euclidean space, whereas groups define the abstract concept of a binary operation along with the addi ...
, the Hilbert space also admits underlying smooth and analytic structures. A vector ξ in ''H'' is said to be smooth or analytic if the map ''g'' → π(''g'') ξ is smooth or analytic (in the norm or weak topologies on ''H''). Smooth vectors are dense in ''H'' by a classical argument of
Lars Gårding
Lars Gårding (7 March 1919 – 7 July 2014) was a Swedish mathematician. He made notable contributions to the study of partial differential equations and partial differential operators. He was a professor of mathematics at Lund University in ...
, since convolution by smooth functions of
compact support
In mathematics, the support of a real-valued function f is the subset of the function domain containing the elements which are not mapped to zero. If the domain of f is a topological space, then the support of f is instead defined as the smalle ...
yields smooth vectors. Analytic vectors are dense by a classical argument of
Edward Nelson
Edward Nelson (May 4, 1932 – September 10, 2014) was an American mathematician. He was professor in the Mathematics Department at Princeton University. He was known for his work on mathematical physics and mathematical logic. In mathematic ...
, amplified by Roe Goodman, since vectors in the image of a heat operator ''e''
–tD, corresponding to an
elliptic differential operator
In the theory of partial differential equations, elliptic operators are differential operators that generalize the Laplace operator. They are defined by the condition that the coefficients of the highest-order derivatives be positive, which imp ...
''D'' in the
universal enveloping algebra
In mathematics, the universal enveloping algebra of a Lie algebra is the unital associative algebra whose representations correspond precisely to the representations of that Lie algebra.
Universal enveloping algebras are used in the represent ...
of ''G'', are analytic. Not only do smooth or analytic vectors form dense subspaces; they also form common cores for the unbounded skew-adjoint operators corresponding to the elements of the
Lie algebra
In mathematics, a Lie algebra (pronounced ) is a vector space \mathfrak g together with an operation called the Lie bracket, an alternating bilinear map \mathfrak g \times \mathfrak g \rightarrow \mathfrak g, that satisfies the Jacobi identi ...
, in the sense of
spectral theory In mathematics, spectral theory is an inclusive term for theories extending the eigenvector and eigenvalue theory of a single square matrix to a much broader theory of the structure of operators in a variety of mathematical spaces. It is a result ...
.
Two unitary representations π
1: ''G'' → U(''H''
1), π
2: ''G'' → U(''H''
2) are said to be unitarily equivalent if there is a
unitary transformation
In mathematics, a unitary transformation is a transformation that preserves the inner product: the inner product of two vectors before the transformation is equal to their inner product after the transformation.
Formal definition
More precisely, ...
''A'':''H''
1 → ''H''
2 such that π
1(''g'') = ''A''
* ∘ π
2(''g'') ∘ ''A'' for all ''g'' in ''G''. When this holds, ''A'' is said to be an
intertwining operator
In mathematics, equivariance is a form of symmetry for functions from one space with symmetry to another (such as symmetric spaces). A function is said to be an equivariant map when its domain and codomain are acted on by the same symmetry grou ...
for the representations
.
If
is a representation of a connected Lie group
on a ''finite-dimensional'' Hilbert space
, then
is unitary if and only if the associated Lie algebra representation
maps into the space of skew-self-adjoint operators on
.
Complete reducibility
A unitary representation is
completely reducible, in the sense that for any closed
invariant subspace In mathematics, an invariant subspace of a linear mapping ''T'' : ''V'' → ''V '' i.e. from some vector space ''V'' to itself, is a subspace ''W'' of ''V'' that is preserved by ''T''; that is, ''T''(''W'') ⊆ ''W''.
General desc ...
, the
orthogonal complement In the mathematical fields of linear algebra and functional analysis, the orthogonal complement of a subspace ''W'' of a vector space ''V'' equipped with a bilinear form ''B'' is the set ''W''⊥ of all vectors in ''V'' that are orthogonal to every ...
is again a closed invariant subspace. This is at the level of an observation, but is a fundamental property. For example, it implies that finite-dimensional unitary representations are always a direct sum of irreducible representations, in the algebraic sense.
Since unitary representations are much easier to handle than the general case, it is natural to consider unitarizable representations, those that become unitary on the introduction of a suitable complex Hilbert space structure. This works very well for
finite group
Finite is the opposite of infinite. It may refer to:
* Finite number (disambiguation)
* Finite set, a set whose cardinality (number of elements) is some natural number
* Finite verb, a verb form that has a subject, usually being inflected or ma ...
s, and more generally for
compact group
In mathematics, a compact (topological) group is a topological group whose topology realizes it as a compact topological space (when an element of the group is operated on, the result is also within the group). Compact groups are a natural gen ...
s, by an averaging argument applied to an arbitrary hermitian structure.
[ Section 4.4] For example, a natural proof of
Maschke's theorem
In mathematics, Maschke's theorem, named after Heinrich Maschke, is a theorem in group representation theory that concerns the decomposition of representations of a finite group into irreducible pieces. Maschke's theorem allows one to make gener ...
is by this route.
Unitarizability and the unitary dual question
In general, for non-compact groups, it is a more serious question which representations are unitarizable. One of the important unsolved problems in mathematics is the description of the unitary dual, the effective classification of irreducible unitary representations of all real
reductive Lie group
In mathematics, a Lie group (pronounced ) is a group that is also a differentiable manifold. A manifold is a space that locally resembles Euclidean space, whereas groups define the abstract concept of a binary operation along with the addi ...
s. All
irreducible unitary representations are
admissible (or rather their
Harish-Chandra modules are), and the admissible representations are given by the
Langlands classification In mathematics, the Langlands classification is a description of the irreducible representations of a reductive Lie group ''G'', suggested by Robert Langlands (1973). There are two slightly different versions of the Langlands classification. One ...
, and it is easy to tell which of them have a non-trivial invariant
sesquilinear form
In mathematics, a sesquilinear form is a generalization of a bilinear form that, in turn, is a generalization of the concept of the dot product of Euclidean space. A bilinear form is linear in each of its arguments, but a sesquilinear form allows o ...
. The problem is that it is in general hard to tell when the quadratic form is
positive definite In mathematics, positive definiteness is a property of any object to which a bilinear form or a sesquilinear form may be naturally associated, which is positive-definite. See, in particular:
* Positive-definite bilinear form
* Positive-definite fu ...
. For many reductive Lie groups this has been solved; see
representation theory of SL2(R) and
representation theory of the Lorentz group
The Lorentz group is a Lie group of symmetries of the spacetime of special relativity. This group can be realized as a collection of matrices, linear transformations, or unitary operators on some Hilbert space; it has a variety of representati ...
for examples.
Notes
References
*
*
*{{citation, title=Harmonic Analysis on Semi-simple Lie Groups I, first=Garth, last= Warner, year=1972, publisher=Springer-Verlag, isbn=0-387-05468-5
See also
*
Induced representations
*
Isotypical representation
*
Representation theory of SL2(R)
*
Representations of the Lorentz group
*
Stone–von Neumann theorem In mathematics and in theoretical physics, the Stone–von Neumann theorem refers to any one of a number of different formulations of the uniqueness of the canonical commutation relations between position and momentum operators. It is named afte ...
*
Unitary representation of a star Lie superalgebra
*
Zonal spherical function In mathematics, a zonal spherical function or often just spherical function is a function on a locally compact group ''G'' with compact subgroup ''K'' (often a maximal compact subgroup) that arises as the matrix coefficient of a ''K''-invariant v ...