“Projective representation”的意思、由来-开放百科全书

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; 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:

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 representations

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

which is the quotient by the subgroup {{math|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 {{math|ρ: G → PGL(V)}} cannot be lifted to a linear representation {{math|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 {{math|G}} to a linear representation of a different group {{math|H}}, which will be a central extension of {{math|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

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

by setting

. The ordinary representation

is a lift of the projective representation

is the sense that:

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

Group cohomology

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

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

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

cohomologous to {{math|c}}. Thus {{math|L}} defines a unique class in {{math|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 {{math|G}}. If {{math|G}} is correctly extended we obtain a linear representation of the extended group, which induces the original projective representation when pushed back down to {{math|G}}. The solution is always a central extension. From Schur's lemma, it follows that the irreducible representations of central extensions of {{math|G}}, and the irreducible projective representations of {{math|G}}, are essentially the same objects.

First example: discrete Fourier transform

Consider the field

of integers mod

, where

is prime, and let

be the

-dimensional space of functions on

with values in

. For each

, define two operators,

and

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

, the operators

and

commute up to multiplication by a constant:

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

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 groups

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

. (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

, and define a projective representation

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)

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 representations

Finite-dimensional projective unitary representations

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

. 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

.^[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

, 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 case

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

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 theorem

On the other hand, Bargmann's theorem states that if the two-dimensional Lie algebra cohomology

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

. 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

, 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

by setting

. Then

is a lift of

in the sense that

, where

is the quotient map from

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

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

must be just a direct product of the universal cover of

with a copy of the real line. We can then lift

from

(by composing with the covering map) and finally restrict this lift to the universal cover of

Notes

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

References