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

PGL(V) = GL(V) / F,

where GL(V) is the general linear group of invertible linear transformations of V over F, and F is the normal subgroup consisting of nonzero scalar multiples of the identity; scalar transformations).[1]

In more concrete terms, a projective representation is a collection of operators , where it is understood that each is only defined up to multiplication by a constant. These should satisfy the homomorphism property up to a constant:

for some constants .

Since each is only defined up to a constant anyway, it does not strictly speaking make sense to ask whether the constants are equal to 1. Nevertheless, one can ask whether it is possible to choose a particular representative of each family of operators in such a way that the 's satisfy the homomorphism property on the nose, not just up to a constant. If such a choice is possible, we say that can be "de-projectivized," or that can be "lifted to an ordinary representation." This possibility is discussed further below.

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Linear representations and projective representationsEdit

One way in which a projective representation can arise is by taking a linear group representation of G on V and applying the quotient map

 

which is the quotient by the subgroup F of scalar transformations (diagonal matrices with all diagonal entries equal). The interest for algebra is in the process in the other direction: given a projective representation, try to 'lift' it to an ordinary linear representation. A general projective representation ρ: G → PGL(V) cannot be lifted to a linear representation G → GL(V), and the obstruction to this lifting can be understood via group homology, as described below.

However, one can lift a projective representation   of G to a linear representation of a different group H, which will be a central extension of G. The group   is the subgroup of   defined as follows:

 ,

where   is the quotient map of   onto  . Since   is a homomorphism, it is easy to check that   is, indeed, a subgroup of  . If the original projective representation   is faithful, then   is isomorphic to the preimage in   of  .

We can define a homomorphism   by setting  . The kernel of   is:

 ,

which is contained in the center of  . It is clear also that   is surjective, so that   is a central extension of  . We can also define an ordinary representation   of   by setting  . The ordinary representation   of   is a lift of the projective representation   of   is the sense that:

 .

If G is a perfect group there is a single universal perfect central extension of G that can be used.

Group cohomologyEdit

The analysis of the lifting question involves group cohomology. Indeed, if one fixes for each g in G a lifted element L(g) in lifting from PGL(V) back to GL(V), the lifts then satisfy

 

for some scalar c(g,h) in F. It follows that the 2-cocycle or Schur multiplier c satisfies the cocycle equation

 

for all g, h, k in G. This c depends on the choice of the lift L; a different choice of lift L′(g) = f(g) L(g) will result in a different cocycle

 

cohomologous to c. Thus L defines a unique class in H2(G, F). This class might not be trivial. For example, in the case of the symmetric group and alternating 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 an extension problem for G. If G is correctly extended we obtain a linear representation of the extended group, which induces the original projective representation when pushed back down to G. The solution is always a central extension. From Schur's lemma, it follows that the irreducible representations of central extensions of G, and the irreducible projective representations of G, are essentially the same objects.

First example: discrete Fourier transformEdit

Consider the field   of integers mod  , where   is prime, and let   be the  -dimensional space of functions on   with values in  . For each   in  , define two operators,   and   on   as follows:

 
 .

We write the formula for   as if   and   were integers, but it is easily seen that the result only depends on the value of   and   mod  . The operator   is a translation, while   is a shift in frequency space (that is, it has the effect of translating the discrete Fourier transform of  ).

One may easily verify that for any   and   in  , the operators   and   commute up to multiplication by a constant:

 .

We may therefore define a projective representation   of   as follows:

 ,

where   denotes the image of an operator   in the quotient group  . Since   and   commute up to a constant,   is easily seen to be a projective representation. On the other hand, since   and   do not actually commute—and no nonzero multiples of them will commute—  cannot be lifted to an ordinary (linear) representation of  .

Since the projective representation   is faithful, the central extension   of   obtained by the construction in the previous section is just the preimage in   of the image of  . Explicitly, this means that   is the group of all operators of the form

 

for  . This group is a discrete version of the Heisenberg group and is isomorphic to the group of matrices of the form

 

with  .

Projective representations of Lie groupsEdit

Studying projective representations of Lie groups leads one to consider true representations of their central extensions (see Group extension#Lie groups). In many cases of interest it suffices to consider representations of covering groups. Specifically, suppose   is a connected cover of a connected Lie group  , so that   for a discrete central subgroup   of  . (Note that   is a special sort of central extension of  .) Suppose also that   is an irreducible unitary representation of   (possibly infinite dimensional). Then by Schur's lemma, the central subgroup   will act by scalar multiples of the identity. Thus, at the projective level,   will descend to  . That is to say, for each  , we can choose a preimage   of   in  , and define a projective representation   of   by setting

 ,

where   denotes the image in   of an operator  . Since   is contained in the center of   and the center of   acts as scalars, the value of   does not depend on the choice of  .

The preceding construction is an important source of examples of projective representations. Bargmann's theorem (discussed below) gives a criterion under which every irreducible projective unitary representation of   arises in this way.

Projective representations of SO(3)Edit

A physically important example of the above construction comes from the case of the rotation group SO(3), whose universal cover is SU(2). According to the representation theory of SU(2), there is exactly one irreducible representation of SU(2) in each dimension. When the dimension is odd (the "integer spin" case), the representation descends to an ordinary representation of SO(3).[3] When the dimension is even (the "fractional spin" case), the representation does not descend to an ordinary representation of SO(3) but does (by the result discussed above) descend to a projective representation of SO(3). Such projective representations of SO(3) (the ones that do not come from ordinary representations) are referred to as "spinorial representations."

By an argument discussed below, every finite-dimensional, irreducible projective representation of SO(3) comes from a finite-dimensional, irreducible ordinary representation of SU(2).

Examples of covers, leading to projective representationsEdit

Notable cases of covering groups giving interesting projective representations:

Finite-dimensional projective unitary representationsEdit

In quantum physics, symmetry of a physical system is typically implemented by means of a projective unitary representation   of a Lie group   on the quantum Hilbert space, that is, a continuous homomorphism

 ,

where   is the quotient of the unitary group   by the operators of the form  . The reason for taking the quotient is that physically, two vectors in the Hilbert space that differ by a constant represent the same physical state. [That is to say, the space of (pure) states is the set of equivalence classes of unit vectors that differ by a constant.] 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   then gives rise to a projective unitary representation   of the Lie algebra   of  . In the finite-dimensional case, it is always possible to "de-projectivize" the Lie-algebra representation   simply by choosing a representative for each   having trace zero.[4] In light of the homomorphisms theorem, it is then possible to de-projectivize   itself, but at the expense of passing to the universal cover   of  .[5] That is to say, every finite-dimensional projective unitary representation of   arises from an ordinary unitary representation of   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   arises from a determinant-one ordinary unitary representation of   (i.e., one in which each element of   acts as an operator with determinant one). If   is semisimple, then every element of   is a linear combination of commutators, in which case every representation of   is by operators with trace zero. In the semisimple case, then, the associated linear representation of   is unique.

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

An important example is the case of SO(3), whose universal cover is SU(2). Now, the Lie algebra   is semisimple. Furthermore, since SU(2) is a compact group, every finite-dimensional representation of it admits an inner product with respect to which the representation is unitary.[6] Thus, the irreducible projective representations of SO(3) are in one-to-one correspondence with the irreducible ordinary representations of SU(2).

Infinite-dimensional projective unitary representations: The Heisenberg caseEdit

The results of the previous subsection do not hold in the infinite-dimensional case, simply because the trace of   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  , acting on the Hilbert space  .[7] These operators are defined as follows:

 
 ,

for all  . These operators are simply continuous versions of the operators   and   described in the "First example" section above. As in that section, we can then define a projective unitary representation   of  :

 ,

because the operators commute up to a phase 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 the Heisenberg group, which is a one-dimensional central extension of  .[8] (See also the Stone–von Neumann theorem.)

Infinite-dimensional projective unitary representations: Bargmann's theoremEdit

On the other hand, Bargmann's theorem states that if the two-dimensional Lie algebra cohomology   of   is trivial, then every projective unitary representation of   can be de-projectivized after passing to the universal cover.[9][10] More precisely, suppose we begin with a projective unitary representation   of a Lie group  . Then the theorem states that   can be lifted to an ordinary unitary representation   of the universal cover   of  . This means that   maps each element of the kernel of the covering map to a scalar multiple of the identity—so that at the projective level,   descends to  —and that the associated projective representation of   is equal to  .

The theorem does not apply to the group  —as the previous example shows—because the two-dimensional cohomology of the associated commutative Lie algebra is nontrivial. Examples where the result does apply include semisimple groups (e.g., SL(2,R)) and the Poincaré group. This last result is important for Wigner's classification of the projective unitary representations of the Poincaré group.

The proof of Bargmann's theorem goes by considering a central extension   of  , constructed similarly to the section above on linear representations and projective representations, as a subgroup of the direct product group  , where   is the Hilbert space on which   acts and   is the group of unitary operators on  . The group   is defined as

 .

As in the earlier section, the map   given by   is a surjective homomorphism whose kernel is   so that   is a central extension of  . Again as in the earlier section, we can then define a linear representation   of   by setting  . Then   is a lift of   in the sense that  , where   is the quotient map from   to  .

A key technical point is to show that   is a Lie group. (This claim is not so obvious, because if   is infinite dimensional, the group   is an infinite-dimensional topological group.) Once this result is established, we see that   is a one-dimensional Lie group central extension of  , so that the Lie algebra   of   is also a one-dimensional central extension of  . But the cohomology group   may be identified with the space of one-dimensional central extensions of  ; if   is trivial then every one-dimensional central extension of   is trivial. In that case,   is just the direct sum of   with a copy of the real line. It follows that the universal cover   of   must be just a direct product of the universal cover of   with a copy of the real line. We can then lift   from   to   (by composing with the covering map) and finally restrict this lift to the universal cover of  .

NotesEdit

  1. ^ Gannon 2006, pp. 176–179.
  2. ^ Schur 1911
  3. ^ Hall 2015 Section 4.7
  4. ^ Hall 2013 Proposition 16.46
  5. ^ Hall 2013 Theorem 16.47
  6. ^ Hall 2015 proof of Theorem 4.28
  7. ^ Hall 2013 Example 16.56
  8. ^ Hall 2013 Exercise 6 in Chapter 14
  9. ^ Bargmann 1954
  10. ^ Simms 1971

ReferencesEdit

  • Bargmann, Valentine (1954), "On unitary ray representations of continuous groups", Annals of Mathematics, 59: 1–46
  • 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, 267, Springer, ISBN 978-1461471158
  • Hall, Brian C. (2015), Lie Groups, Lie Algebras, and Representations: An Elementary Introduction, Graduate Texts in Mathematics, 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: 283–287

See alsoEdit