representation theory of Lie groups
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In mathematics and
theoretical physics Theoretical physics is a branch of physics that employs mathematical models and abstractions of physical objects and systems to rationalize, explain and predict natural phenomena. This is in contrast to experimental physics, which uses experim ...
, a representation of a Lie group is a linear action of a Lie group on a
vector space In mathematics and physics, a vector space (also called a linear space) is a set whose elements, often called '' vectors'', may be added together and multiplied ("scaled") by numbers called ''scalars''. Scalars are often real numbers, but can ...
. Equivalently, a representation is a smooth homomorphism of the group into the group of invertible operators on the vector space. Representations play an important role in the study of continuous symmetry. A great deal is known about such representations, a basic tool in their study being the use of the corresponding 'infinitesimal'
representations of Lie algebras In the mathematical field of representation theory, a Lie algebra representation or representation of a Lie algebra is a way of writing a Lie algebra as a set of matrices (or endomorphisms of a vector space) in such a way that the Lie bracket is ...
.


Finite-dimensional representations


Representations

A complex
representation of a group In the mathematical field of representation theory, group representations describe abstract groups in terms of bijective linear transformations of a vector space to itself (i.e. vector space automorphisms); in particular, they can be used to re ...
is an action by a group on a finite-dimensional vector space over the field \mathbb C. A representation of the Lie group ''G'', acting on an ''n''-dimensional
vector space In mathematics and physics, a vector space (also called a linear space) is a set whose elements, often called '' vectors'', may be added together and multiplied ("scaled") by numbers called ''scalars''. Scalars are often real numbers, but can ...
''V'' over \mathbb C is then a smooth
group homomorphism In mathematics, given two groups, (''G'', ∗) and (''H'', ·), a group homomorphism from (''G'', ∗) to (''H'', ·) is a function ''h'' : ''G'' → ''H'' such that for all ''u'' and ''v'' in ''G'' it holds that : h(u*v) = h(u) \cdot h(v) w ...
:\Pi:G\rightarrow\operatorname(V), where \operatorname(V) is the
general linear group In mathematics, the general linear group of degree ''n'' is the set of invertible matrices, together with the operation of ordinary matrix multiplication. This forms a group, because the product of two invertible matrices is again invertible, ...
of all invertible linear transformations of V under their composition. Since all ''n''-dimensional spaces are isomorphic, the group \operatorname(V) can be identified with the group of the invertible, complex generally Smoothness of the map \Pi can be regarded as a technicality, in that any continuous homomorphism will automatically be smooth. We can alternatively describe a representation of a Lie group G as a ''linear action'' of G on a vector space V. Notationally, we would then write g\cdot v in place of \Pi(g)v for the way a group element g\in G acts on the vector v\in V. A typical example in which representations arise in physics would be the study of a linear partial differential equation having symmetry group G. Although the individual solutions of the equation may not be invariant under the action of G, the ''space'' V of all solutions is invariant under the action of G. Thus, V constitutes a representation of G. See the example of SO(3), discussed below.


Basic definitions

If the homomorphism \Pi is injective (i.e., a
monomorphism In the context of abstract algebra or universal algebra, a monomorphism is an injective homomorphism. A monomorphism from to is often denoted with the notation X\hookrightarrow Y. In the more general setting of category theory, a monomorphism ...
), the representation is said to be faithful. If a
basis Basis may refer to: Finance and accounting * Adjusted basis, the net cost of an asset after adjusting for various tax-related items *Basis point, 0.01%, often used in the context of interest rates * Basis trading, a trading strategy consisting ...
for the complex vector space ''V'' is chosen, the representation can be expressed as a homomorphism into
general linear group In mathematics, the general linear group of degree ''n'' is the set of invertible matrices, together with the operation of ordinary matrix multiplication. This forms a group, because the product of two invertible matrices is again invertible, ...
\operatorname(n;\mathbb C). This is known as a ''matrix representation''. Two representations of ''G'' on vector spaces ''V'', ''W'' are ''equivalent'' if they have the same matrix representations with respect to some choices of bases for ''V'' and ''W''. Given a representation \Pi:G\rightarrow\operatorname(V), we say that a subspace ''W'' of ''V'' is an invariant subspace if \Pi(g)w\in W for all g\in G and w\in W. The representation is said to be irreducible if the only invariant subspaces of ''V'' are the zero space and ''V'' itself. For certain types of Lie groups, namely compact Theorem 4.28 and semisimple groups, every finite-dimensional representation decomposes as a direct sum of irreducible representations, a property known as complete reducibility. For such groups, a typical goal of representation theory is to classify all finite-dimensional irreducible representations of the given group, up to isomorphism. (See the Classification section below.) A
unitary representation In mathematics, a unitary representation of a group ''G'' is a linear representation π of ''G'' on a complex Hilbert space ''V'' such that π(''g'') is a unitary operator for every ''g'' ∈ ''G''. The general theory is well-developed in case ''G ...
on a finite-dimensional inner product space is defined in the same way, except that \Pi is required to map into the group of
unitary operators In functional analysis Functional analysis is a branch of mathematical analysis, the core of which is formed by the study of vector spaces endowed with some kind of limit-related structure (e.g. inner product, norm, topology, etc.) and ...
. If ''G'' is a
compact Lie 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 ...
, every finite-dimensional representation is equivalent to a unitary one.


Lie algebra representations

Each representation of a Lie group ''G'' gives rise to a representation of its Lie algebra; this correspondence is discussed in detail in subsequent sections. See representation of Lie algebras for the Lie algebra theory.


An example: The rotation group SO(3)

In quantum mechanics, the time-independent
Schrödinger equation The Schrödinger equation is a linear partial differential equation that governs the wave function of a quantum-mechanical system. It is a key result in quantum mechanics, and its discovery was a significant landmark in the development of th ...
, \hat\psi=E\psi plays an important role. In the three-dimensional case, if \hat has rotational symmetry, then the space V_E of solutions to \hat\psi=E\psi will be invariant under the action of SO(3). Thus, V_E will—for each fixed value of E—constitute a representation of SO(3), which is typically finite dimensional. In trying to solve \hat\psi=E\psi, it helps to know what all possible finite-dimensional representations of SO(3) look like. The representation theory of SO(3) plays a key role, for example, in the mathematical analysis of the hydrogen atom. Every standard textbook on quantum mechanics contains an analysis which essentially classifies finite-dimensional irreducible representations of SO(3), by means of its Lie algebra. (The commutation relations among the angular momentum operators are just the relations for the Lie algebra \mathfrak(3) of SO(3).) One subtlety of this analysis is that the representations of the group and the Lie algebra are not in one-to-one correspondence, a point that is critical in understanding the distinction between
integer spin In particle physics, a boson ( ) is a subatomic particle whose spin quantum number has an integer value (0,1,2 ...). Bosons form one of the two fundamental classes of subatomic particle, the other being fermions, which have odd half-integer spi ...
and half-integer spin.


Ordinary representations

The
rotation group SO(3) In mechanics and geometry, the 3D rotation group, often denoted SO(3), is the group of all rotations about the origin of three-dimensional Euclidean space \R^3 under the operation of composition. By definition, a rotation about the origin is a ...
is a compact Lie group and thus every finite-dimensional representation of SO(3) decomposes as a direct sum of irreducible representations. The group SO(3) has one irreducible representation in each odd dimension. For each non-negative integer k, the irreducible representation of dimension 2k+1 can be realized as the space V_k of homogeneous harmonic polynomials on \mathbb^3 of degree k. Here, SO(3) acts on V_k in the usual way that rotations act on functions on \mathbb^3: :(\Pi(R)f)(x)=f(R^x)\quad R\in \operatorname(3). The restriction to the unit sphere S^2 of the elements of V_k are the
spherical harmonics In mathematics and physical science, spherical harmonics are special functions defined on the surface of a sphere. They are often employed in solving partial differential equations in many scientific fields. Since the spherical harmonics form ...
of degree k. If, say, k=1, then all polynomials that are homogeneous of degree one are harmonic, and we obtain a three-dimensional space V_1 spanned by the linear polynomials x, y, and z. If k=2, the space V_2 is spanned by the polynomials xy, xz, yz, x^2-y^2, and x^2-z^2. As noted above, the finite-dimensional representations of SO(3) arise naturally when studying the time-independent Schrödinger equation for a radial potential, such as the hydrogen atom, as a reflection of the rotational symmetry of the problem. (See the role played by the spherical harmonics in the mathematical analysis of hydrogen.)


Projective representations

If we look at the Lie algebra \mathfrak(3) of SO(3), this Lie algebra is isomorphic to the Lie algebra \mathfrak(2) of SU(2). By the representation theory of \mathfrak(2), there is then one irreducible representation of \mathfrak(3) in ''every'' dimension. The even-dimensional representations, however, do not correspond to representations of the ''group'' SO(3). These so-called "fractional spin" representations do, however, correspond to ''projective'' representations of SO(3). These representations arise in the quantum mechanics of particles with fractional spin, such as an electron.


Operations on representations

In this section, we describe three basic operations on representations. See also the corresponding constructions for representations of a Lie algebra.


Direct sums

If we have two representations of a group G, \Pi_1:G\rightarrow GL(V_1) and \Pi_2:G\rightarrow GL(V_2), then the direct sum would have V_1\oplus V_2 as the underlying vector space, with the action of the group given by :\Pi(g)(v_1,v_2)=(\Pi_1(g)v_1,\Pi_2(g)v_2), for all v_1\in V_1, v_2\in V_2, and g\in G. Certain types of Lie groups—notably, compact Lie groups—have the property that ''every'' finite-dimensional representation is isomorphic to a direct sum of irreducible representations. In such cases, the classification of representations reduces to the classification of irreducible representations. See
Weyl's theorem on complete reducibility In algebra, Weyl's theorem on complete reducibility is a fundamental result in the theory of Lie algebra representations (specifically in the representation theory of semisimple Lie algebras). Let \mathfrak be a semisimple Lie algebra over a field ...
.


Tensor products of representations

If we have two representations of a group G, \Pi_1:G\rightarrow GL(V_1) and \Pi_2:G\rightarrow GL(V_2), then the tensor product of the representations would have the
tensor product In mathematics, the tensor product V \otimes W of two vector spaces and (over the same field) is a vector space to which is associated a bilinear map V\times W \to V\otimes W that maps a pair (v,w),\ v\in V, w\in W to an element of V \otime ...
vector space V_1\otimes V_2 as the underlying vector space, with the action of G uniquely determined by the assumption that :\Pi(g)(v_1\otimes v_2)=(\Pi_1(g)v_1)\otimes(\Pi_2(g)v_2) for all v_1\in V_1 and v_2\in V_2. That is to say, \Pi(g)=\Pi_1(g)\otimes\Pi_2(g). The Lie algebra representation \pi associated to the tensor product representation \Pi is given by the formula: :\pi(X)=\pi_1(X)\otimes I+I\otimes\pi_2(X). The tensor product of two irreducible representations is usually not irreducible; a basic problem in representation theory is then to decompose tensor products of irreducible representations as a direct sum of irreducible subspaces. This problem goes under the name of "addition of angular momentum" or " Clebsch–Gordan theory" in the physics literature.


Dual representations

Let G be a Lie group and \Pi:G\rightarrow GL(V) be a representation of G. Let V^* be the dual space, that is, the space of linear functionals on V. Then we can define a representation \Pi^*:G\rightarrow GL(V^*) by the formula :\Pi^*(g)=(\Pi(g^))^\operatorname, where for any operator A:V\rightarrow V, the transpose operator A^\operatorname:V^*\rightarrow V^* is defined as the "composition with A" operator: :(A^\operatorname\phi)(v)=\phi(Av). (If we work in a basis, then A^ is just the usual matrix transpose of A.) The inverse in the definition of \Pi^* is needed to ensure that \Pi^* is actually a representation of G, in light of the identity (AB)^\operatorname=B^\operatornameA^\operatorname. The dual of an irreducible representation is always irreducible, but may or may not be isomorphic to the original representation. In the case of the group SU(3), for example, the
irreducible representations In mathematics, specifically in the representation theory of groups and algebras, an irreducible representation (\rho, V) or irrep of an algebraic structure A is a nonzero representation that has no proper nontrivial subrepresentation (\rho, _W,W ...
are labeled by a pair (m_1,m_2) of non-negative integers. The dual of the representation associated to (m_1,m_2) is the representation associated to (m_2,m_1).


Lie group versus Lie algebra representations


Overview

In many cases, it is convenient to study representations of a Lie group by studying representations of the associated Lie algebra. In general, however, not every representation of the Lie algebra comes from a representation of the group. This fact is, for example, lying behind the distinction between
integer spin In particle physics, a boson ( ) is a subatomic particle whose spin quantum number has an integer value (0,1,2 ...). Bosons form one of the two fundamental classes of subatomic particle, the other being fermions, which have odd half-integer spi ...
and half-integer spin in quantum mechanics. On the other hand, if ''G'' is a simply connected group, then a theorem says that we do, in fact, get a one-to-one correspondence between the group and Lie algebra representations. Let be a Lie group with Lie algebra \mathfrak g, and assume that a representation \pi of \mathfrak g is at hand. The
Lie correspondence A lie is an assertion that is believed to be false, typically used with the purpose of deceiving or misleading someone. The practice of communicating lies is called lying. A person who communicates a lie may be termed a liar. Lies can be int ...
may be employed for obtaining group representations of the connected component of the . Roughly speaking, this is effected by taking the
matrix exponential In mathematics, the matrix exponential is a matrix function on square matrices analogous to the ordinary exponential function. It is used to solve systems of linear differential equations. In the theory of Lie groups, the matrix exponential give ...
of the matrices of the Lie algebra representation. A subtlety arises if is not simply connected. This may result in
projective representation In the field of representation theory in mathematics, a projective representation of a group ''G'' on a vector space ''V'' over a field ''F'' is a group homomorphism from ''G'' to the projective linear group \mathrm(V) = \mathrm(V) / F^*, where G ...
s or, in physics parlance, multi-valued representations of . These are actually representations of the universal covering group of . These results will be explained more fully below. The Lie correspondence gives results only for the connected component of the groups, and thus the other components of the full group are treated separately by giving representatives for matrices representing these components, one for each component. These form (representatives of) the zeroth homotopy group of . For example, in the case of the four-component
Lorentz group In physics and mathematics, the Lorentz group is the group of all Lorentz transformations of Minkowski spacetime, the classical and quantum setting for all (non-gravitational) physical phenomena. The Lorentz group is named for the Dutch physicis ...
, representatives of space inversion and time reversal must be put in by hand. Further illustrations will be drawn from the
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 ...
below.


The exponential mapping

If G is a Lie group with Lie algebra \mathfrak g, then we have the exponential map from \mathfrak g to G, written as :X\mapsto e^X,\quad X\in\mathfrak g. If G is a matrix Lie group, the expression e^X can be computed by the usual power series for the exponential. In any Lie group, there exist neighborhoods U of the identity in G and V of the origin in \mathfrak g with the property that every g in U can be written uniquely as g=e^X with X\in V. That is, the exponential map has a ''local'' inverse. In most groups, this is only local; that is, the exponential map is typically neither one-to-one nor onto.


Lie algebra representations from group representations

It is always possible to pass from a representation of a Lie group to a representation of its Lie algebra \mathfrak. If is a group representation for some vector space , then its
pushforward The notion of pushforward in mathematics is "dual" to the notion of pullback, and can mean a number of different but closely related things. * Pushforward (differential), the differential of a smooth map between manifolds, and the "pushforward" op ...
(differential) at the identity, or Lie map, \pi : \mathfrak \to \text V is a Lie algebra representation. It is explicitly computed using, Theorem 3.28 A basic property relating \Pi and \pi involves the exponential map: :\Pi(e^X)=e^. The question we wish to investigate is whether every representation of \mathfrak g arises in this way from representations of the group G. As we shall see, this is the case when G is simply connected.


Group representations from Lie algebra representations

The main result of this section is the following: :Theorem: If G is simply connected, then every representation \pi of the Lie algebra \mathfrak g of G comes from a representation \Pi of G itself. From this we easily deduce the following: :Corollary: If G is connected but not simply connected, every representation \pi of \mathfrak g comes from a representation \Pi of \tilde G, the universal cover of G. If \pi is irreducible, then \Pi descends to a ''
projective representation In the field of representation theory in mathematics, a projective representation of a group ''G'' on a vector space ''V'' over a field ''F'' is a group homomorphism from ''G'' to the projective linear group \mathrm(V) = \mathrm(V) / F^*, where G ...
'' of G. A projective representation is one in which each \Pi(g),\,g\in G, is defined only up to multiplication by a constant. In quantum physics, it is natural to allow projective representations in addition to ordinary ones, because states are really defined only up to a constant. (That is to say, if \psi is a vector in the quantum Hilbert space, then c\psi represents the same physical state for any constant c.) Every ''finite-dimensional'' projective representation of a connected Lie group G comes from an ordinary representation of the universal cover \tilde G of G. Conversely, as we will discuss below, every irreducible ordinary representation of \tilde G descends to a projective representation of G. In the physics literature, projective representations are often described as multi-valued representations (i.e., each \Pi(g) does not have a single value but a whole family of values). This phenomenon is important to the study of fractional spin in quantum mechanics. We now outline the proof of the main results above. Suppose \pi : \mathfrak \to \mathfrak(V) is a representation of \mathfrak g on a vector space . If there is going to be an associated Lie group representation \Pi, it must satisfy the exponential relation of the previous subsection. Now, in light of the local invertibility of the exponential, we can ''define'' a map \Pi from a neighborhood U of the identity in G by this relation: :\Pi(e^X)=e^,\quad g=e^X\in U. A key question is then this: Is this locally defined map a "local homomorphism"? (This question would apply even in the special case where the exponential mapping is globally one-to-one and onto; in that case, \Pi would be a globally defined map, but it is not obvious why \Pi would be a homomorphism.) The answer to this question is yes: \Pi is a local homomorphism, and this can be established using the Baker–Campbell–Hausdorff formula. If G is connected, then every element of G is at least a ''product'' of exponentials of elements of \mathfrak g. Thus, we can tentatively define \Pi globally as follows. Note, however, that the representation of a given group element as a product of exponentials is very far from unique, so it is very far from clear that \Pi is actually well defined. To address the question of whether \Pi is well defined, we connect each group element g\in G to the identity using a continuous path. It is then possible to define \Pi along the path, and to show that the value of \Pi(g) is unchanged under continuous deformation of the path with endpoints fixed. If G is simply connected, any path starting at the identity and ending at g can be continuously deformed into any other such path, showing that \Pi(g) is fully independent of the choice of path. Given that the initial definition of \Pi near the identity was a local homomorphism, it is not difficult to show that the globally defined map is also a homomorphism satisfying . If G is not simply connected, we may apply the above procedure to the universal cover \tilde G of G. Let p:\tilde G\rightarrow G be the covering map. If it should happen that the kernel of \Pi:\tilde G\rightarrow \operatorname(V) contains the kernel of p, then \Pi descends to a representation of the original group G. Even if this is not the case, note that the kernel of p is a discrete normal subgroup of \tilde G, which is therefore in the center of \tilde G. Thus, if \pi is irreducible,
Schur's lemma In mathematics, Schur's lemma is an elementary but extremely useful statement in representation theory of groups and algebras. In the group case it says that if ''M'' and ''N'' are two finite-dimensional irreducible representations of a group ' ...
implies that the kernel of p will act by scalar multiples of the identity. Thus, \Pi descends to a ''projective'' representation of G, that is, one that is defined only modulo scalar multiples of the identity. A pictorial view of how the universal covering group contains ''all'' such homotopy classes, and a technical definition of it (as a set and as a group) is given in geometric view. For example, when this is specialized to the doubly connected , the universal covering group is \text(2,\Complex), and whether its corresponding representation is faithful decides whether is projective.


Classification in the compact case

If ''G'' is a connected
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 British ...
Lie group, its finite-dimensional representations can be decomposed as direct sums of irreducible representations. The irreducibles are classified by a "
theorem of the highest weight In representation theory, a branch of mathematics, the theorem of the highest weight classifies the irreducible representations of a complex semisimple Lie algebra \mathfrak g. Theorems 9.4 and 9.5 There is a closely related theorem classifying the ...
." We give a brief description of this theory here; for more details, see the articles on representation theory of a connected compact Lie group and the parallel theory classifying representations of semisimple Lie algebras. Let ''T'' be a
maximal torus In the mathematical theory of compact Lie groups a special role is played by torus subgroups, in particular by the maximal torus subgroups. A torus in a compact Lie group ''G'' is a compact, connected, abelian Lie subgroup of ''G'' (and therefor ...
in ''G''. By
Schur's lemma In mathematics, Schur's lemma is an elementary but extremely useful statement in representation theory of groups and algebras. In the group case it says that if ''M'' and ''N'' are two finite-dimensional irreducible representations of a group ' ...
, the irreducible representations of ''T'' are one dimensional. These representations can be classified easily and are labeled by certain "analytically integral elements" or "weights." If \Sigma is an irreducible representation of ''G'', the restriction of \Sigma to ''T'' will usually not be irreducible, but it will decompose as a direct sum of irreducible representations of ''T'', labeled by the associated weights. (The same weight can occur more than once.) For a fixed \Sigma, one can identify one of the weights as "highest" and the representations are then classified by this highest weight. An important aspect of the representation theory is the associated theory of
characters Character or Characters may refer to: Arts, entertainment, and media Literature * ''Character'' (novel), a 1936 Dutch novel by Ferdinand Bordewijk * ''Characters'' (Theophrastus), a classical Greek set of character sketches attributed to The ...
. Here, for a representation \Sigma of ''G'', the character is the function :\chi_G:G\rightarrow\mathbb given by :\chi_G(g)=\operatorname(\Sigma(g)). Two representations with the same character turn out to be isomorphic. Furthermore, the
Weyl character formula In mathematics, the Weyl character formula in representation theory describes the characters of irreducible representations of compact Lie groups in terms of their highest weights. It was proved by . There is a closely related formula for the ch ...
gives a remarkable formula for the character of a representation in terms of its highest weight. Not only does this formula gives a lot of useful information about the representation, but it plays a crucial role in the proof of the theorem of the highest weight.


Unitary representations on Hilbert spaces

Let ''V'' be a complex Hilbert space, which may be infinite dimensional, and let U(V) denote the group of unitary operators on ''V''. A unitary representation of a Lie group ''G'' on ''V'' is a
group homomorphism In mathematics, given two groups, (''G'', ∗) and (''H'', ·), a group homomorphism from (''G'', ∗) to (''H'', ·) is a function ''h'' : ''G'' → ''H'' such that for all ''u'' and ''v'' in ''G'' it holds that : h(u*v) = h(u) \cdot h(v) w ...
\Pi:G\rightarrow U(V) with the property that for each fixed v\in V, the map :g\mapsto \Pi(g)v is a continuous map of ''G'' into ''V''.


Finite-dimensional unitary representations

If the Hilbert space ''V'' is finite-dimensional, there is an associated representation \pi of the Lie algebra \mathfrak g of G. If G is connected, then the representation \Pi of G is unitary if and only if \pi(X) is skew-self-adjoint for each X\in\mathfrak g. If G is
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 British ...
, then every representation \Pi of G on a finite-dimensional vector space ''V'' is "unitarizable," meaning that it is possible to choose an inner product on ''V'' so that each \Pi(g),\, g\in G is unitary. proof of Proposition 4.28


Infinite-dimensional unitary representations

If the Hilbert space ''V'' is allowed to be infinite dimensional, the study of unitary representations involves a number of interesting features that are not present in the finite dimensional case. For example, the construction of an appropriate representation of the Lie algebra \mathfrak g becomes technically challenging. One setting in which the Lie algebra representation is well understood is that of
semisimple In mathematics, semi-simplicity is a widespread concept in disciplines such as linear algebra, abstract algebra, representation theory, category theory, and algebraic geometry. A semi-simple object is one that can be decomposed into a sum of ''sim ...
(or reductive) Lie groups, where the associated Lie algebra representation forms a
(g,K)-module In mathematics, more specifically in the representation theory of reductive Lie groups, a (\mathfrak,K)-module is an algebraic object, first introduced by Harish-Chandra, used to deal with continuous infinite-dimensional representations using algeb ...
. Examples of unitary representations arise in quantum mechanics and quantum field theory, but also in Fourier analysis as shown in the following example. Let G=\mathbb R, and let the complex Hilbert space ''V'' be L^2(\mathbb R). We define the representation \psi:\mathbb R\rightarrow U(L^2(\mathbb R)) by : psi(a)(f)x)=f(x-a). Here are some important examples in which unitary representations of a Lie group have been analyzed. * The Stone–von Neumann theorem can be understood as giving a classification of the irreducible unitary representations of the
Heisenberg group In mathematics, the Heisenberg group H, named after Werner Heisenberg, is the group of 3×3 upper triangular matrices of the form ::\begin 1 & a & c\\ 0 & 1 & b\\ 0 & 0 & 1\\ \end under the operation of matrix multiplication. Elements ...
. *
Wigner's classification In mathematics and theoretical physics, Wigner's classification is a classification of the nonnegative ~ (~E \ge 0~)~ energy irreducible unitary representations of the Poincaré group which have either finite or zero mass eigenvalues. (Since thi ...
for representations of the
Poincaré group The Poincaré group, named after Henri Poincaré (1906), was first defined by Hermann Minkowski (1908) as the group of Minkowski spacetime isometries. It is a ten-dimensional non-abelian Lie group that is of importance as a model in our und ...
plays a major conceptual role in quantum field theory by showing how the mass and spin of particles can be understood in group-theoretic terms. * The representation theory of SL(2,R) was worked out by V. Bargmann and serves as the prototype for the study of unitary representations of noncompact semisimple Lie groups.


Projective representations

In quantum physics, one is often interested in ''projective'' unitary representations of a Lie group G. The reason for this interest is that states of a quantum system are represented by vectors in a Hilbert space \mathbf H—but with the understanding that two states differing by a constant are actually the same physical state. The symmetries of the Hilbert space are then described by unitary operators, but a unitary operator that is a multiple of the identity does not change the physical state of the system. Thus, we are interested not in ordinary unitary representations—that is, homomorphisms of G into the unitary group U(\mathbf H)—but rather in projective unitary representations—that is, homomorphisms of G into the projective unitary group :PU(\mathbf H):=U(\mathbf H)/\. To put it differently, for a projective representation, we construct a family of unitary operators \rho(g),\,\, g\in G, where it is understood that changing \rho(g) by a constant of absolute value 1 is counted as "the same" operator. The operators \rho(g) are then required to satisfy the homomorphism property ''up to a constant'': :\rho(g)\rho(h)=e^\rho(gh). We have already discussed the irreducible projective unitary representations of the rotation group SO(3) above; considering projective representations allows for fractional spin in addition to integer spin. Bargmann's theorem states that for certain types of Lie groups G, irreducible projective unitary representations of G are in one-to-one correspondence with ordinary unitary representations of the universal cover of G. Important examples where Bargmann's theorem applies are SO(3) (as just mentioned) and the
Poincaré group The Poincaré group, named after Henri Poincaré (1906), was first defined by Hermann Minkowski (1908) as the group of Minkowski spacetime isometries. It is a ten-dimensional non-abelian Lie group that is of importance as a model in our und ...
. The latter case is important to
Wigner's classification In mathematics and theoretical physics, Wigner's classification is a classification of the nonnegative ~ (~E \ge 0~)~ energy irreducible unitary representations of the Poincaré group which have either finite or zero mass eigenvalues. (Since thi ...
of the projective representations of the Poincaré group, with applications to quantum field theory. One example where Bargmann's theorem does ''not'' apply is the group \mathbb R^. The set of translations in position and momentum on L^2(\mathbb R^n) form a projective unitary representation of \mathbb R^ but they do not come from an ordinary representation of the universal cover of \mathbb R^—which is just \mathbb R^ itself. In this case, to get an ordinary representation, one has to pass to the
Heisenberg group In mathematics, the Heisenberg group H, named after Werner Heisenberg, is the group of 3×3 upper triangular matrices of the form ::\begin 1 & a & c\\ 0 & 1 & b\\ 0 & 0 & 1\\ \end under the operation of matrix multiplication. Elements ...
, which is a one-dimensional central extension of \mathbb R^. (See the discussion
here Here is an adverb that means "in, on, or at this place". It may also refer to: Software * Here Technologies, a mapping company * Here WeGo (formerly Here Maps), a mobile app and map website by Here Technologies, Here Television * Here TV (form ...
.)


The commutative case

If G is a commutative Lie group, then every irreducible unitary representation of G on complex vector spaces is one dimensional. (This claim follows from
Schur's lemma In mathematics, Schur's lemma is an elementary but extremely useful statement in representation theory of groups and algebras. In the group case it says that if ''M'' and ''N'' are two finite-dimensional irreducible representations of a group ' ...
and holds even if the representations are not assumed ahead of time to be finite dimensional.) Thus, the irreducible unitary representations of G are simply continuous homomorphisms of G into the unit circle group, U(1). For example, if G=\mathbb R, the irreducible unitary representations have the form :\Pi(x)= ^/math>, for some real number a. See also
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 ...
for this case.


See also

* Representation theory of connected compact groups * Lie algebra representation *
Projective representation In the field of representation theory in mathematics, a projective representation of a group ''G'' on a vector space ''V'' over a field ''F'' is a group homomorphism from ''G'' to the projective linear group \mathrm(V) = \mathrm(V) / F^*, where G ...
*
Representation theory of SU(2) In the study of the representation theory of Lie groups, the study of representations of SU(2) is fundamental to the study of representations of semisimple Lie groups. It is the first case of a Lie group that is both a compact group and a non-abel ...
*
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 ...
* Representation theory of Hopf algebras *
Adjoint representation of a Lie group In mathematics, the adjoint representation (or adjoint action) of a Lie group ''G'' is a way of representing the elements of the group as linear transformations of the group's Lie algebra, considered as a vector space. For example, if ''G'' is GL( ...
* List of Lie group topics *
Symmetry in quantum mechanics Symmetries in quantum mechanics describe features of spacetime and particles which are unchanged under some transformation, in the context of quantum mechanics, relativistic quantum mechanics and quantum field theory, and with applications in the ...
*
Wigner D-matrix The Wigner D-matrix is a unitary matrix in an irreducible representation of the groups SU(2) and SO(3). It was introduced in 1927 by Eugene Wigner, and plays a fundamental role in the quantum mechanical theory of angular momentum. The complex con ...


Notes


References

* * . * . * . * . The 2003 reprint corrects several typographical mistakes. * {{Authority control Lie groups