In the field ofrepresentation theory inmathematics, aprojective representation of agroupG on avector spaceV over afieldF is agroup homomorphism fromG to theprojective linear group$${\displaystyle \mathrm{P}\mathrm{G}\mathrm{L}(V)=\mathrm{G}\mathrm{L}(V)/F^{\ast },}$$where GL(V) is thegeneral linear group of invertiblelinear transformations ofV overF, andF^∗ is thenormal subgroup consisting of nonzero scalar multiples of the identity transformation (seeScalar transformation).^[1]

Just as linear representations study the possible actions of the groupG on vector spaces via linear transformation, the projective representations study the actions on lines in these vector spaces (namelyV /F^*) via linear transformations.

In more concrete terms, a projective representation of${\displaystyle G}$ can be represented as a collection of operators${\displaystyle \rho (g)\in \mathrm{G}\mathrm{L}(V),g\in G}$ satisfying the homomorphism property up to a constant:

${\displaystyle \rho (g)\rho (h)=c(g,h)\rho (gh),}$

for some constant${\displaystyle c(g,h)\in F}$. Two such choices of operators${\displaystyle {\rho }_1,{\rho }_2}$ define the same projective representation if for any${\displaystyle g\in G}$ the choices${\displaystyle {\rho }_1(g),{\rho }_2(g)}$ are the same up to a scalar. Equivalently, a projective representation of${\displaystyle G}$ is a collection of operators${\displaystyle \tilde{\rho }(g)\subset \mathrm{G}\mathrm{L}(V),g\in G}$, such that${\displaystyle \tilde{\rho }(gh)=\tilde{\rho }(g)\tilde{\rho }(h)}$. Note that, in this notation,${\displaystyle \tilde{\rho }(g)}$ is aset of linear operators related by multiplication with some nonzero scalar.

If it is possible to choose a particular representative${\displaystyle \rho (g)\in \tilde{\rho }(g)}$ in each family of operators in such a way that the homomorphism property is satisfiedexactly, rather than just up to a constant, then we say that${\displaystyle \tilde{\rho }}$ can be "de-projectivized", or that${\displaystyle \tilde{\rho }}$ can be "lifted to an ordinary representation". More concretely, we thus say that${\displaystyle \tilde{\rho }}$ can be de-projectivized if there are${\displaystyle \rho (g)\in \tilde{\rho }(g)}$ for each${\displaystyle g\in G}$ such that${\displaystyle \rho (g)\rho (h)=\rho (gh)}$. This possibility is discussed further below.

One way in which a projective representation can arise is by taking a lineargroup representation ofG onV and applying the quotient map

${\displaystyle \mathrm{GL}(V,F)\to \mathrm{PGL}(V,F)}$

which is the quotient by the subgroupF^∗ ofscalar transformations (diagonal matrices with all diagonal entries equal). The interest for algebra is in the process in the other direction: given aprojective representation, try to 'lift' it to an ordinarylinear representation. A general projective representationρ:G → PGL(V) cannot be lifted to a linear representationG → GL(V), and theobstruction to this lifting can be understood viagroup cohomology, as described below.

However, onecan lift a projective representation${\displaystyle \rho }$ ofG to a linear representation of a different groupH, which will be acentral extension ofG. The group${\displaystyle H}$ is the subgroup of${\displaystyle G\times \mathrm{G}\mathrm{L}(V)}$ defined as follows:

${\displaystyle H=\{(g,A)\in G\times \mathrm{G}\mathrm{L}(V)\mid \pi (A)=\rho (g)\}}$,

where${\displaystyle \pi }$ is the quotient map of${\displaystyle \mathrm{G}\mathrm{L}(V)}$ onto${\displaystyle \mathrm{P}\mathrm{G}\mathrm{L}(V)}$. Since${\displaystyle \rho }$ is a homomorphism, it is easy to check that${\displaystyle H}$ is, indeed, a subgroup of${\displaystyle G\times \mathrm{G}\mathrm{L}(V)}$. If the original projective representation${\displaystyle \rho }$ is faithful, then${\displaystyle H}$ is isomorphic to the preimage in${\displaystyle \mathrm{G}\mathrm{L}(V)}$ of${\displaystyle \rho (G)\subseteq \mathrm{P}\mathrm{G}\mathrm{L}(V)}$.

We can define a homomorphism${\displaystyle \varphi :H\to G}$ by setting${\displaystyle \varphi ((g,A))=g}$. The kernel of${\displaystyle \varphi }$ is:

${\displaystyle \mathrm{k}\mathrm{e}\mathrm{r}(\varphi )=\{(e,cI)\mid c\in F^{\ast }\}}$,

which is contained in the center of${\displaystyle H}$. It is clear also that${\displaystyle \varphi }$ is surjective, so that${\displaystyle H}$ is a central extension of${\displaystyle G}$. We can also define an ordinary representation${\displaystyle \sigma }$ of${\displaystyle H}$ by setting${\displaystyle \sigma ((g,A))=A}$. Theordinary representation${\displaystyle \sigma }$ of${\displaystyle H}$ is a lift of theprojective representation${\displaystyle \rho }$ of${\displaystyle G}$ in the sense that:

${\displaystyle \pi (\sigma ((g,A)))=\rho (g)=\rho (\varphi ((g,A)))}$.

IfG is aperfect group there is a singleuniversal perfect central extension ofG that can be used.

The analysis of the lifting question involvesgroup cohomology. Indeed, if one fixes for eachg inG a lifted elementL(g) in lifting fromPGL(V) back toGL(V), the lifts then satisfy

${\displaystyle L(gh)=c(g,h)L(g)L(h)}$

for some scalarc(g,h) inF^∗. It follows that the 2-cocycle orSchur multiplierc satisfies the cocycle equation

${\displaystyle c(h,k)c(g,hk)=c(g,h)c(gh,k)}$

for allg,h,k inG. Thisc depends on the choice of the liftL; a different choice of liftL′(g) =f(g)L(g) will result in a different cocycle

${\displaystyle c^{\mathrm{\prime }}(g,h)=f(gh)f(g{)}^{-1}f(h{)}^{-1}c(g,h)}$

cohomologous toc. ThusL defines a unique class inH²(G,F^∗). This class might not be trivial. For example, in the case of thesymmetric group andalternating group, Schur established that there is exactly one non-trivial class of Schur multiplier, and completely determined all the corresponding irreducible representations.^[2]

In general, a nontrivial class leads to anextension problem forG. IfG is correctly extended we obtain a linear representation of the extended group, which induces the original projective representation when pushed back down toG. The solution is always acentral extension. FromSchur's lemma, it follows that theirreducible representations of central extensions ofG, and the irreducible projective representations ofG, are essentially the same objects.

Consider first the group${\displaystyle G=\mathbb{Z}/2}$ with two elements denoted by${\displaystyle e,\sigma }$, where${\displaystyle e}$ is the trivial element and the group is defined by the identity${\displaystyle {\sigma }^2=e}$. In every projective representation${\displaystyle e}$ is sent to the identity map, so the representation is determined completely by the image of${\displaystyle \sigma }$. This group has two (irreducible)linear representations, both 1-dimensional:

• Thetrivial representation:${\displaystyle \sigma \mapsto 1}$
• The nontrivial representation:${\displaystyle \sigma \mapsto -1}$

While these are distinct as linear representations, namely their actions on the points are different, they are the same as projective representations. In a 1-dimensional space, there is a single line, and both send that line to itself. More generally, any choice of the image${\displaystyle \sigma \mapsto c}$ will produce the same 1-dimensional projective representation. Namely, there is a unique 1-dimensional projective representation, and it is the trivial representation (and this holds for any group).

Over the 2-dimensional plane${\displaystyle {\mathbb{R}}^2}$, we can send${\displaystyle \sigma }$ to the${\displaystyle {90}^{\circ }}$ rotation:

Two${\displaystyle {90}^{\circ }}$ rotations, namely${\displaystyle {180}^{\circ }}$ rotation, returns the lines in the plane to themselves, hence it defines a projective representation. More formally,${\displaystyle \rho ({\sigma }^2),\rho (\sigma {)}^2}$ are the same up to a sign:

$${\displaystyle \rho (\sigma {)}^2=-Id=-\rho (e)=-\rho ({\sigma }^2).}$$

Unlike the 1-dimensional case, this projective representation doesn't arise from a standard linear representation. However, working over the complex numbers${\displaystyle \mathbb{C}}$ instead of the real numbers, this projective representation is the same as

in which case it is a linear representation, since${\displaystyle {\rho }^{\prime }(\sigma {)}^2={\rho }^{\prime }({\sigma }^2)=Id}$.

Now let${\displaystyle G=\mathbb{Z}/n}$ which we write multiplicatively as${\displaystyle G=\{e,\sigma ,{\sigma }^2,...,{\sigma }^{n-1}\}}$, and let${\displaystyle \rho :G\to GL(V)}$ be a choice of representatives for a projective representation:

• The trivial element: Since${\displaystyle \rho (e)\rho (e)=c(e,e)\rho (e^2)=c(e,e)\rho (e)}$, and${\displaystyle \rho (e)}$ is invertible, it must be the scalar${\displaystyle \rho (e)=c(e,e)\cdot Id}$. Hence, we can change the choice to be${\displaystyle \rho (e)=Id}$ without changing the representation.
• The nontrivial elements: Since${\displaystyle \rho (\sigma {)}^k,\rho ({\sigma }^k)}$ are the same up to a scalar, we can similarly choose the representatives so that${\displaystyle \rho ({\sigma }^k)=\rho (\sigma {)}^k}$ for${\displaystyle k=0,...,n-1}$.

After these reductions, we are left with a choice for${\displaystyle \rho (\sigma )}$ and a scalar${\displaystyle c}$ such that

$${\displaystyle \rho (\sigma {)}^n=c\cdot \rho ({\sigma }^n)=c\cdot \rho (e)=c\cdot Id.}$$

On the other hand, any choice of a root${\displaystyle A^n=c\cdot Id}$ for some scalar${\displaystyle c\ne 0}$ can define a projective representation by setting$${\displaystyle \rho ({\sigma }^k)=A^k,k=0,...,n-1.}$$

Over the complex numbers (or generally overalgebraically closed field), changing the choice of${\displaystyle \rho (\sigma )}$ into${\displaystyle \frac{1}{\sqrt[n]{c}}\rho (\sigma )}$ will produce a standard linear representation, or in other words all the projective representations arise from ordinary linear representations

In a product of two cyclic groups${\displaystyle G=\{e,\sigma ,...{\sigma }^{n-1}\}\times \{e,\tau ,...,{\tau }^{m-1}\}}$, a similar process can be done by changing the choice of representatives to satisfy:

This reduces to three conditions:

1. ${\displaystyle \rho (\sigma {)}^n=c_{\sigma }\cdot Id}$ ,${\displaystyle \rho (\tau {)}^m=c_{\tau }\cdot Id}$, and
2. ${\displaystyle \rho (\tau )\rho (\sigma )=c_{\tau ,\sigma }\rho (\sigma \tau )=c_{\tau ,\sigma }\rho (\sigma )\rho (\tau )}$.

This new scalar must satisfy:

• ${\displaystyle c_{\tau }\rho (\sigma )=\rho (\tau {)}^m\rho (\sigma )=c_{\tau ,\sigma }^m\rho (\sigma )\rho (\tau {)}^m=c_{\tau }{c}_{\tau ,\sigma }^m\rho (\sigma )}$, so that${\displaystyle c_{\tau ,\sigma }^m=1}$.
• Similarly,${\displaystyle c_{\tau ,\sigma }^n=1}$.

Both together imply that${\displaystyle c_{\tau ,\sigma }^{\text{gcd}(n,m)}=1}$. Any such choices of${\displaystyle \rho (\sigma ),\rho (\tau )}$ which satisfy these 3 conditions will define a projective representation.

Over the complex numbers, as previously we can assume that${\displaystyle c_{\tau }=c_{\sigma }=1}$, so the only parameter is${\displaystyle c_{\tau ,\sigma }^{\text{gcd}(n,m)}=1}$.

For example, taking${\displaystyle n=m}$ we have 3 conditions${\displaystyle \rho (\sigma {)}^n=\rho (\tau {)}^n=Id}$ and${\displaystyle \rho (\tau )\rho (\sigma )=c_{\tau ,\sigma }\rho (\sigma )\rho (\tau )}$ where${\displaystyle c_{\tau ,\sigma }^n=1}$. These equations have solutions in${\displaystyle G{L}_n(\mathbb{C})}$ for any${\displaystyle n}$ and any choice of root of unity${\displaystyle c_{\tau ,\sigma }^n=1}$. More specifically, letting${\displaystyle {\zeta }_n}$ be such aroot of unity, define:

.

Both matrices define an "${\displaystyle n}$-rotation":${\displaystyle \rho (\sigma )}$ rotates each coordinate separately as complex numbers, while${\displaystyle \rho (\tau )}$ rotates the coordinates of the vector, and in particular${\displaystyle \rho (\sigma {)}^n=\rho (\tau {)}^n=Id}$. In addition we have that${\displaystyle \rho (\tau )\rho (\sigma )={\zeta }_n\rho (\sigma )\rho (\tau )}$.

Note that${\displaystyle c_{\tau ,\sigma }=\rho (\tau )\rho (\sigma )\rho (\tau {)}^{-1}\rho (\sigma {)}^{-1}}$ is independent of the choices of representatives. It follows that the${\displaystyle n}$ projective representations defined above for the${\displaystyle n}$ distinct roots of unity are distinct representations.

The projective representations for general product of cyclic groups is done in a similar manner.

The projective representations of${\displaystyle \mathbb{Z}/p\times \mathbb{Z}/p}$ as mentioned above, are many times viewed through the lens of Fourier transform.

Consider the field${\displaystyle \mathbb{Z}/p}$ of integers mod${\displaystyle p}$, where${\displaystyle p}$ is prime, and let${\displaystyle V}$ be the${\displaystyle p}$-dimensional space of functions on${\displaystyle \mathbb{Z}/p}$ with values in${\displaystyle \mathbb{C}}$. For each${\displaystyle a}$ in${\displaystyle \mathbb{Z}/p}$, define two operators,${\displaystyle T_a}$ and${\displaystyle S_a}$ on${\displaystyle V}$ as follows:

We write the formula for${\displaystyle S_a}$ as if${\displaystyle a}$ and${\displaystyle b}$ were integers, but it is easily seen that the result only depends on the value of${\displaystyle a}$ and${\displaystyle b}$ mod${\displaystyle p}$. The operator${\displaystyle T_a}$ is a translation, while${\displaystyle S_a}$ is a shift in frequency space (that is, it has the effect of translating thediscrete Fourier transform of${\displaystyle f}$).

One may easily verify that for any${\displaystyle a}$ and${\displaystyle b}$ in${\displaystyle \mathbb{Z}/p}$, the operators${\displaystyle T_a}$ and${\displaystyle S_b}$ commute up to multiplication by a constant:

${\displaystyle T_a{S}_b=e^{-2\pi iab/p}{S}_b{T}_a}$.

We may therefore define a projective representation${\displaystyle \rho }$ of${\displaystyle \mathbb{Z}/p\times \mathbb{Z}/p}$ as follows:

${\displaystyle \rho (a,b)=[T_a{S}_b]}$,

where${\displaystyle [A]}$ denotes the image of an operator${\displaystyle A\in \mathrm{G}\mathrm{L}(V)}$ in the quotient group${\displaystyle \mathrm{P}\mathrm{G}\mathrm{L}(V)}$. Since${\displaystyle T_a}$ and${\displaystyle S_b}$ commute up to a constant,${\displaystyle \rho }$ is easily seen to be a projective representation. On the other hand, since${\displaystyle T_a}$ and${\displaystyle S_b}$ do not actually commute—and no nonzero multiples of them will commute—${\displaystyle \rho }$ cannot be lifted to an ordinary (linear) representation of${\displaystyle \mathbb{Z}/p\times \mathbb{Z}/p}$.

Since the projective representation${\displaystyle \rho }$ is faithful, the central extension${\displaystyle H}$ of${\displaystyle \mathbb{Z}/p\times \mathbb{Z}/p}$ obtained by the construction in the previous section is just the preimage in${\displaystyle \mathrm{G}\mathrm{L}(V)}$ of the image of${\displaystyle \rho }$. Explicitly, this means that${\displaystyle H}$ is the group of all operators of the form

${\displaystyle e^{2\pi ic/p}{T}_a{S}_b}$

for${\displaystyle a,b,c\in \mathbb{Z}/p}$. This group is a discrete version of theHeisenberg group and is isomorphic to the group of matrices of the form

with${\displaystyle a,b,c\in \mathbb{Z}/p}$.

Studying projective representations ofLie groups leads one to consider true representations of their central extensions (seeGroup extension § Lie groups). In many cases of interest it suffices to consider representations ofcovering groups. Specifically, suppose${\displaystyle \hat{G}}$ is a connected cover of a connected Lie group${\displaystyle G}$, so that${\displaystyle G\cong \hat{G}/N}$ for a discrete central subgroup${\displaystyle N}$ of${\displaystyle \hat{G}}$. (Note that${\displaystyle \hat{G}}$ is a special sort of central extension of${\displaystyle G}$.) Suppose also that${\displaystyle \mathrm{\Pi }}$ is an irreducible unitary representation of${\displaystyle \hat{G}}$ (possibly infinite dimensional). Then bySchur's lemma, the central subgroup${\displaystyle N}$ will act by scalar multiples of the identity. Thus, at the projective level,${\displaystyle \mathrm{\Pi }}$ will descend to${\displaystyle G}$. That is to say, for each${\displaystyle g\in G}$, we can choose a preimage${\displaystyle \hat{g}}$ of${\displaystyle g}$ in${\displaystyle \hat{G}}$, and define a projective representation${\displaystyle \rho }$ of${\displaystyle G}$ by setting

${\displaystyle \rho (g)=\left[\mathrm{\Pi }\left(\hat{g}\right)\right]}$,

where${\displaystyle [A]}$ denotes the image in${\displaystyle \mathrm{P}\mathrm{G}\mathrm{L}(V)}$ of an operator${\displaystyle A\in \mathrm{G}\mathrm{L}(V)}$. Since${\displaystyle N}$ is contained in the center of${\displaystyle \hat{G}}$ and the center of${\displaystyle \hat{G}}$acts as scalars, the value of${\displaystyle \left[\mathrm{\Pi }\left(\hat{g}\right)\right]}$ does not depend on the choice of${\displaystyle \hat{g}}$.

The preceding construction is an important source of examples of projective representations. Bargmann's theorem (discussed below) gives a criterion under whichevery irreducible projective unitary representation of${\displaystyle G}$ arises in this way.

A physically important example of the above construction comes from the case of therotation groupSO(3), whoseuniversal cover isSU(2). According to therepresentation theory ofSU(2), there is exactly one irreducible representation ofSU(2) in each dimension. When the dimension is odd (the "integer spin" case), the representation descends to an ordinary representation ofSO(3).^[3] When the dimension is even (the "fractional spin" case), the representation does not descend to an ordinary representation ofSO(3) but does (by the result discussed above) descend to a projective representation ofSO(3). Such projective representations ofSO(3) (the ones that do not come from ordinary representations) are referred to as "spinorial representations", whose elements (vectors) are calledspinors.

By an argument discussed below, every finite-dimensional, irreducibleprojective representation ofSO(3) comes from a finite-dimensional, irreducibleordinary representation ofSU(2).

Notable cases of covering groups giving interesting projective representations:

• Thespecial orthogonal groupSO(n,F) is doubly covered by theSpin groupSpin(n,F).
• In particular, thegroupSO(3) (the rotation group in 3 dimensions) is doubly covered bySU(2). This has important applications in quantum mechanics, as thestudy of representations ofSU(2) leads to a nonrelativistic (low-velocity) theory ofspin.
• The groupSO⁺(3;1), isomorphic to theMöbius group, is likewise doubly covered bySL₂(C). Both are supergroups of aforementionedSO(3) andSU(2) respectively and form arelativistic spin theory.
• The universal cover of thePoincaré group is a double cover (thesemidirect product ofSL₂(C) withR⁴). The irreducible unitary representations of this cover give rise to projective representations of the Poincaré group, as inWigner's classification. Passing to the cover is essential, in order to include the fractional spin case.
• Theorthogonal groupO(n) is double covered by thePin groupPin_±(n).
• Thesymplectic groupSp(2n)=Sp(2n,R) (not to be confused with the compact real form of the symplectic group, sometimes also denoted bySp(m)) is double covered by themetaplectic groupMp(2n). An important projective representation ofSp(2n) comes from themetaplectic representation ofMp(2n).

In quantum physics,symmetry of a physical system is typically implemented by means of a projective unitary representation${\displaystyle \rho }$ of a Lie group${\displaystyle G}$ on the quantumHilbert space, that is, a continuous homomorphism

${\displaystyle \rho :G\to \mathrm{P}\mathrm{U}(\mathcal{H}),}$

where${\displaystyle \mathrm{P}\mathrm{U}(\mathcal{H})}$ is the quotient of the unitary group${\displaystyle \mathrm{U}(\mathcal{H})}$ by the operators of the form${\displaystyle cI,|c|=1}$. The reason for taking the quotient is that physically, two vectors in the Hilbert space that are proportional represent the same physical state. [That is to say, the space of (pure) states is theset of equivalence classes of unit vectors, where two unit vectors are considered equivalent if they are proportional.] Thus, a unitary operator that is a multiple of the identity actually acts as the identity on the level of physical states.

A finite-dimensional projective representation of${\displaystyle G}$ then gives rise to a projective unitary representation${\displaystyle {\rho }_{\ast }}$ of the Lie algebra${\displaystyle \mathfrak{g}}$ of${\displaystyle G}$. In the finite-dimensional case, it is always possible to "de-projectivize" the Lie-algebra representation${\displaystyle {\rho }_{\ast }}$ simply by choosing a representative for each${\displaystyle {\rho }_{\ast }(X)}$ having trace zero.^[4] In light of thehomomorphisms theorem, it is then possible to de-projectivize${\displaystyle \rho }$ itself, but at the expense of passing to the universal cover${\displaystyle \tilde{G}}$ of${\displaystyle G}$.^[5] That is to say, every finite-dimensional projective unitary representation of${\displaystyle G}$ arises from an ordinary unitary representation of${\displaystyle \tilde{G}}$ by the procedure mentioned at the beginning of this section.

Specifically, since the Lie-algebra representation was de-projectivized by choosing a trace-zero representative, every finite-dimensional projective unitary representation of${\displaystyle G}$ arises from adeterminant-one ordinary unitary representation of${\displaystyle \tilde{G}}$ (i.e., one in which each element of${\displaystyle \tilde{G}}$ acts as an operator with determinant one). If${\displaystyle \mathfrak{g}}$ is semisimple, then every element of${\displaystyle \mathfrak{g}}$ is a linear combination of commutators, in which caseevery representation of${\displaystyle \mathfrak{g}}$ is by operators with trace zero. In the semisimple case, then, the associated linear representation of${\displaystyle \tilde{G}}$ is unique.

Conversely, if${\displaystyle \rho }$ is anirreducible unitary representation of the universal cover${\displaystyle \tilde{G}}$ of${\displaystyle G}$, then bySchur's lemma, the center of${\displaystyle \tilde{G}}$ acts as scalar multiples of the identity. Thus, at the projective level,${\displaystyle \rho }$ descends to a projective representation of the original group${\displaystyle G}$. Thus, there is a natural one-to-one correspondence between the irreducible projective representations of${\displaystyle G}$ and the irreducible, determinant-one ordinary representations of${\displaystyle \tilde{G}}$. (In the semisimple case, the qualifier "determinant-one" may be omitted, because in that case, every representation of${\displaystyle \tilde{G}}$ is automatically determinant one.)

An important example is the case ofSO(3), whose universal cover isSU(2). Now, the Lie algebra${\displaystyle \mathrm{s}\mathrm{u}(2)}$ is semisimple. Furthermore, since SU(2) is acompact group, every finite-dimensional representation of it admits an inner product with respect to which the representation is unitary.^[6] Thus, the irreducibleprojective representations of SO(3) are in one-to-one correspondence with the irreducibleordinary representations of SU(2).

The results of the previous subsection do not hold in the infinite-dimensional case, simply because the trace of${\displaystyle {\rho }_{\ast }(X)}$ is typically not well defined. Indeed, the result fails: Consider, for example, the translations in position space and in momentum space for a quantum particle moving in${\displaystyle {\mathbb{R}}^n}$, acting on the Hilbert space${\displaystyle L^2({\mathbb{R}}^n)}$.^[7] These operators are defined as follows:

for all${\displaystyle a\in {\mathbb{R}}^n}$. These operators are simply continuous versions of the operators${\displaystyle T_a}$ and${\displaystyle S_a}$ described in the "First example" section above. As in that section, we can then define aprojective unitary representation${\displaystyle \rho }$ of${\displaystyle {\mathbb{R}}^{2n}}$:

${\displaystyle \rho (a,b)=[T_a{S}_b],}$

because the operators commute up to aphase factor. But no choice of the phase factors will lead to an ordinary unitary representation, since translations in position do not commute with translations in momentum (and multiplying by a nonzero constant will not change this). These operators do, however, come from an ordinary unitary representation of theHeisenberg group, which is a one-dimensional central extension of${\displaystyle {\mathbb{R}}^{2n}}$.^[8] (See also theStone–von Neumann theorem.)

On the other hand,Bargmann's theorem states that if the secondLie algebra cohomology group${\displaystyle H^2(\mathfrak{g};\mathbb{R})}$ of${\displaystyle \mathfrak{g}}$ is trivial, then every projective unitary representation of${\displaystyle G}$ can be de-projectivized after passing to the universal cover.^[9][10] More precisely, suppose we begin with a projective unitary representation${\displaystyle \rho }$ of a Lie group${\displaystyle G}$. Then the theorem states that${\displaystyle \rho }$ can be lifted to an ordinary unitary representation${\displaystyle \hat{\rho }}$ of the universal cover${\displaystyle \hat{G}}$ of${\displaystyle G}$. This means that${\displaystyle \hat{\rho }}$ maps each element of the kernel of the covering map to a scalar multiple of the identity—so that at the projective level,${\displaystyle \hat{\rho }}$ descends to${\displaystyle G}$—and that the associated projective representation of${\displaystyle G}$ is equal to${\displaystyle \rho }$.

The theorem does not apply to the group${\displaystyle {\mathbb{R}}^{2n}}$—as the previous example shows—because the second cohomology group of the associated commutative Lie algebra is nontrivial. Examples where the result does apply include semisimple groups (e.g.,SL(2,R)) and thePoincaré group. This last result is important forWigner's classification of the projective unitary representations of the Poincaré group.

The proof of Bargmann's theorem goes by considering acentral extension${\displaystyle H}$ of${\displaystyle G}$, constructed similarly to the section above on linear representations and projective representations, as a subgroup of the direct product group${\displaystyle G\times U(\mathcal{H})}$, where${\displaystyle \mathcal{H}}$ is the Hilbert space on which${\displaystyle \rho }$ acts and${\displaystyle U(\mathcal{H})}$ is the group of unitary operators on${\displaystyle \mathcal{H}}$. The group${\displaystyle H}$ is defined as

${\displaystyle H=\{(g,U)\mid \pi (U)=\rho (g)\}.}$

As in the earlier section, the map${\displaystyle \varphi :H\to G}$ given by${\displaystyle \varphi (g,U)=g}$ is a surjective homomorphism whose kernel is${\displaystyle \{(e,cI)\mid |c|=1\},}$ so that${\displaystyle H}$ is a central extension of${\displaystyle G}$. Again as in the earlier section, we can then define a linear representation${\displaystyle \sigma }$ of${\displaystyle H}$ by setting${\displaystyle \sigma (g,U)=U}$. Then${\displaystyle \sigma }$ is a lift of${\displaystyle \rho }$ in the sense that${\displaystyle \rho \circ \varphi =\pi \circ \sigma }$, where${\displaystyle \pi }$ is the quotient map from${\displaystyle U(\mathcal{H})}$ to${\displaystyle PU(\mathcal{H})}$.

A key technical point is to show that${\displaystyle H}$ is aLie group. (This claim is not so obvious, because if${\displaystyle \mathcal{H}}$ is infinite dimensional, the group${\displaystyle G\times U(\mathcal{H})}$ is an infinite-dimensionaltopological group.) Once this result is established, we see that${\displaystyle H}$ is a one-dimensional Lie group central extension of${\displaystyle G}$, so that the Lie algebra${\displaystyle \mathfrak{h}}$ of${\displaystyle H}$ is also a one-dimensional central extension of${\displaystyle \mathfrak{g}}$ (note here that the adjective "one-dimensional" does not refer to${\displaystyle H}$ and${\displaystyle \mathfrak{h}}$, but rather to the kernel of the projection map from those objects onto${\displaystyle G}$ and${\displaystyle \mathfrak{g}}$ respectively). But the cohomology group${\displaystyle H^2(\mathfrak{g};\mathbb{R})}$may be identified with the space of one-dimensional (again, in the aforementioned sense) central extensions of${\displaystyle \mathfrak{g}}$; if${\displaystyle H^2(\mathfrak{g};\mathbb{R})}$ is trivial then every one-dimensional central extension of${\displaystyle \mathfrak{g}}$ is trivial. In that case,${\displaystyle \mathfrak{h}}$ is just the direct sum of${\displaystyle \mathfrak{g}}$ with a copy of the real line. It follows that the universal cover${\displaystyle \tilde{H}}$ of${\displaystyle H}$ must be just a direct product of the universal cover of${\displaystyle G}$ with a copy of the real line. We can then lift${\displaystyle \sigma }$ from${\displaystyle H}$ to${\displaystyle \tilde{H}}$ (by composing with the covering map) and finally restrict this lift to the universal cover${\displaystyle \tilde{G}}$ of${\displaystyle G}$.

• Affine representation
• Group action
• Central extension
• Particle physics and representation theory
• Spin-1/2
• Spinor
• Symmetry in quantum mechanics
• Heisenberg group

• Bargmann, Valentine (1954), "On unitary ray representations of continuous groups",Annals of Mathematics,59 (1):1–46,doi:10.2307/1969831,JSTOR 1969831
• Gannon, Terry (2006),Moonshine Beyond the Monster: The Bridge Connecting Algebra, Modular Forms and Physics, Cambridge University Press,ISBN 978-0-521-83531-2
• Hall, Brian C. (2013),Quantum Theory for Mathematicians, Graduate Texts in Mathematics, vol. 267, Springer,ISBN 978-1461471158
• Hall, Brian C. (2015),Lie Groups, Lie Algebras, and Representations: An Elementary Introduction, Graduate Texts in Mathematics, vol. 222 (2nd ed.), Springer,ISBN 978-3319134666
• Schur, I. (1911),"Über die Darstellung der symmetrischen und der alternierenden Gruppe durch gebrochene lineare Substitutionen",Crelle's Journal,139:155–250
• Simms, D. J. (1971), "A short proof of Bargmann's criterion for the lifting of projective representations of Lie groups",Reports on Mathematical Physics,2 (4):283–287,Bibcode:1971RpMP....2..283S,doi:10.1016/0034-4877(71)90011-5

• Homological algebra
• Group theory
• Representation theory
• Representation theory of groups

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