- 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).
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.
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 in the sense that:
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.
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
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). 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:
- The special orthogonal group SO(n, F) is doubly covered by the Spin group Spin(n, F).
- In particular, the group SO(3) (the rotation group in 3 dimensions) is doubly covered by SU(2). This has important applications in quantum mechanics, as the study of representations of SU(2) leads to a nonrelativistic (low-velocity) theory of spin.
- The group SO+(3;1), isomorphic to the Möbius group, is likewise doubly covered by SL2(C). Both are supergroups of aforementioned SO(3) and SU(2) respectively and form a relativistic spin theory.
- The universal cover of the Poincaré group is a double cover (the semidirect product of SL2(C) with R4). The irreducible unitary representations of this cover give rise to projective representations of the Poincaré group, as in Wigner's classification. Passing to the cover is essential, in order to include the fractional spin case.
- The orthogonal group O(n) is double covered by the Pin group Pin±(n).
- The symplectic group Sp(2n) is double covered by the metaplectic group Mp(2n). An important projective representation of Sp(2n) comes from the metaplectic representation of Mp(2n).
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 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 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. 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 . 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. 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 . 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 . (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. 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 (note here that the adjective "one-dimensional" does not refer to and , but rather to the kernel of the projection map from those objects onto and respectively). But the cohomology group may be identified with the space of one-dimensional (again, in the aforementioned sense) 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 .
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- 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, doi:10.1016/0034-4877(71)90011-5