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
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 transformation (see Scalar transformation).
In more concrete terms, a projective representation is a collection of operators formula_1, where it is understood that each formula_2 is only defined up to multiplication by a constant. These should satisfy the homomorphism property up to a constant:
for some constants formula_4.
Since each formula_2 is only defined up to a constant anyway, it does not strictly speaking make sense to ask whether the constants formula_4 are equal to 1. Nevertheless, one can ask whether it is "possible to choose" a particular representative of each family formula_2 of operators in such a way that the formula_2's satisfy the homomorphism property on the nose, not just up to a constant. If such a choice is possible, we say that formula_9 can be "de-projectivized," or that formula_9 can be "lifted to an ordinary representation." This possibility is discussed further below.
Linear representations and projective representations.
One way in which a projective representation can arise is by taking a linear group representation of on and applying the quotient map
which is the quotient by the subgroup 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 cannot be lifted to a linear representation , and the obstruction to this lifting can be understood via group cohomology, as described below. 
However, one "can" lift a projective representation formula_9 of to a linear representation of a different group , which will be a central extension of . The group formula_13 is the subgroup of formula_14 defined as follows:
where formula_16 is the quotient map of formula_17 onto formula_18. Since formula_9 is a homomorphism, it is easy to check that formula_13 is, indeed, a subgroup of formula_14. If the original projective representation formula_9 is faithful, then formula_13 is isomorphic to the preimage in formula_17 of formula_25.
We can define a homomorphism formula_26 by setting formula_27. The kernel of formula_28 is:
which is contained in the center of formula_13. It is clear also that formula_28 is surjective, so that formula_13 is a central extension of formula_33. We can also define an ordinary representation formula_34 of formula_13 by setting formula_36. The "ordinary" representation formula_34 of formula_13 is a lift of the "projective" representation formula_9 of formula_33 in the sense that:
If is a perfect group there is a single universal perfect central extension of that can be used.
Group cohomology.
The analysis of the lifting question involves group cohomology. Indeed, if one fixes for each in a lifted element in lifting from back to , the lifts then satisfy
for some scalar in . It follows that the 2-cocycle or Schur multiplier satisfies the cocycle equation
for all in . This depends on the choice of the lift ; a different choice of lift will result in a different cocycle
cohomologous to . Thus defines a unique class in . 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.
In general, a nontrivial class leads to an extension problem for . If is correctly extended we obtain a linear representation of the extended group, which induces the original projective representation when pushed back down to . The solution is always a central extension. From Schur's lemma, it follows that the irreducible representations of central extensions of , and the irreducible projective representations of , are essentially the same objects.
First example: discrete Fourier transform.
Consider the field formula_45 of integers mod formula_46, where formula_46 is prime, and let formula_48 be the formula_46-dimensional space of functions on formula_45 with values in formula_51. For each formula_52 in formula_45, define two operators, formula_54 and formula_55 on formula_48 as follows:
We write the formula for formula_55 as if formula_52 and formula_60 were integers, but it is easily seen that the result only depends on the value of formula_52 and formula_60 mod formula_46. The operator formula_54 is a translation, while formula_55 is a shift in frequency space (that is, it has the effect of translating the discrete Fourier transform of formula_66).
One may easily verify that for any formula_52 and formula_60 in formula_45, the operators formula_54 and formula_71 commute up to multiplication by a constant:
We may therefore define a projective representation formula_9 of formula_74 as follows:
where formula_76 denotes the image of an operator formula_77 in the quotient group formula_18. Since formula_54 and formula_71 commute up to a constant, formula_9 is easily seen to be a projective representation. On the other hand, since formula_54 and formula_71 do not actually commute—and no nonzero multiples of them will commute—formula_9 cannot be lifted to an ordinary (linear) representation of formula_74.
Since the projective representation formula_9 is faithful, the central extension formula_13 of formula_74 obtained by the construction in the previous section is just the preimage in formula_17 of the image of formula_9. Explicitly, this means that formula_13 is the group of all operators of the form
for formula_93. This group is a discrete version of the Heisenberg group and is isomorphic to the group of matrices of the form
with formula_95.
Projective representations of Lie groups.
Studying projective representations of Lie groups leads one to consider true representations of their central extensions (see ). In many cases of interest it suffices to consider representations of covering groups. Specifically, suppose formula_96 is a connected cover of a connected Lie group formula_33, so that formula_98 for a discrete central subgroup formula_99 of formula_96. (Note that formula_96 is a special sort of central extension of formula_33.) Suppose also that formula_103 is an irreducible unitary representation of formula_96 (possibly infinite dimensional). Then by Schur's lemma, the central subgroup formula_99 will act by scalar multiples of the identity. Thus, at the projective level, formula_103 will descend to formula_33. That is to say, for each formula_108, we can choose a preimage formula_109 of formula_110 in formula_96, and define a projective representation formula_9 of formula_33 by setting
where formula_76 denotes the image in formula_18 of an operator formula_77. Since formula_99 is contained in the center of formula_96 and the center of formula_96 acts as scalars, the value of formula_121 does not depend on the choice of formula_109.
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 formula_33 arises in this way.
Projective representations of SO(3).
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). 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 representations.
Notable cases of covering groups giving interesting projective representations:
Finite-dimensional projective unitary representations.
In quantum physics, symmetry of a physical system is typically implemented by means of a projective unitary representation formula_9 of a Lie group formula_33 on the quantum Hilbert space, that is, a continuous homomorphism
where formula_127 is the quotient of the unitary group formula_128 by the operators of the form formula_129. 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 the set 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 formula_33 then gives rise to a projective unitary representation formula_131 of the Lie algebra formula_132 of formula_33. In the finite-dimensional case, it is always possible to "de-projectivize" the Lie-algebra representation formula_131 simply by choosing a representative for each formula_135 having trace zero. In light of the homomorphisms theorem, it is then possible to de-projectivize formula_9 itself, but at the expense of passing to the universal cover formula_137 of formula_33. That is to say, every finite-dimensional projective unitary representation of formula_33 arises from an ordinary unitary representation of formula_137 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 formula_33 arises from a "determinant-one" ordinary unitary representation of formula_137 (i.e., one in which each element of formula_137 acts as an operator with determinant one). If formula_132 is semisimple, then every element of formula_132 is a linear combination of commutators, in which case "every" representation of formula_132 is by operators with trace zero. In the semisimple case, then, the associated linear representation of formula_137 is unique.
Conversely, if formula_9 is an "irreducible" unitary representation of the universal cover formula_137 of formula_33, then by Schur's lemma, the center of formula_137 acts as scalar multiples of the identity. Thus, at the projective level, formula_9 descends to a projective representation of the original group formula_33. Thus, there is a natural one-to-one correspondence between the irreducible projective representations of formula_33 and the irreducible, determinant-one ordinary representations of formula_137. (In the semisimple case, the qualifier "determinant-one" may be omitted, because in that case, every representation of formula_137 is automatically determinant one.)
An important example is the case of SO(3), whose universal cover is SU(2). Now, the Lie algebra formula_157 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. 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 case.
The results of the previous subsection do not hold in the infinite-dimensional case, simply because the trace of formula_135 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 formula_159, acting on the Hilbert space formula_160. These operators are defined as follows:
for all formula_162. These operators are simply continuous versions of the operators formula_54 and formula_55 described in the "First example" section above. As in that section, we can then define a "projective" unitary representation formula_9 of formula_166:
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 formula_166. (See also the Stone–von Neumann theorem.)
Infinite-dimensional projective unitary representations: Bargmann's theorem.
On the other hand, Bargmann's theorem states that if the two-dimensional Lie algebra cohomology formula_169 of formula_132 is trivial, then every projective unitary representation of formula_33 can be de-projectivized after passing to the universal cover. More precisely, suppose we begin with a projective unitary representation formula_9 of a Lie group formula_33. Then the theorem states that formula_9 can be lifted to an ordinary unitary representation formula_175 of the universal cover formula_96 of formula_33. This means that formula_175 maps each element of the kernel of the covering map to a scalar multiple of the identity—so that at the projective level, formula_175 descends to formula_33—and that the associated projective representation of formula_33 is equal to formula_9.
The theorem does not apply to the group formula_166—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 formula_13 of formula_33, constructed similarly to the section above on linear representations and projective representations, as a subgroup of the direct product group formula_186, where formula_187 is the Hilbert space on which formula_9 acts and formula_189 is the group of unitary operators on formula_187. The group formula_13 is defined as
As in the earlier section, the map formula_193 given by formula_194 is a surjective homomorphism whose kernel is formula_195 so that formula_13 is a central extension of formula_33. Again as in the earlier section, we can then define a linear representation formula_34 of formula_13 by setting formula_200. Then formula_34 is a lift of formula_9 in the sense that formula_203, where formula_16 is the quotient map from formula_189 to formula_206. 
A key technical point is to show that formula_13 is a "Lie" group. (This claim is not so obvious, because if formula_187 is infinite dimensional, the group formula_186 is an infinite-dimensional topological group.) Once this result is established, we see that formula_13 is a one-dimensional Lie group central extension of formula_33, so that the Lie algebra formula_212 of formula_13 is also a one-dimensional central extension of formula_132 (note here that the adjective "one-dimensional" does not refer to formula_13 and formula_216, but rather to the kernel of the projection map from those objects onto formula_33 and formula_218 respectively). But the cohomology group formula_169 may be identified with the space of one-dimensional (again, in the aforementioned sense) central extensions of formula_132; if formula_169 is trivial then every one-dimensional central extension of formula_132 is trivial. In that case, formula_212 is just the direct sum of formula_132 with a copy of the real line. It follows that the universal cover formula_225 of formula_13 must be just a direct product of the universal cover of formula_33 with a copy of the real line. We can then lift formula_34 from formula_13 to formula_225 (by composing with the covering map) and finally restrict this lift to the universal cover formula_137 of formula_33.