Central Extensions

This section, along with the one following it, largely follows the arguments of Weinberg in [78]. Suppose there is a projective representation U of a Lie group, with general elementsg parametrized by a finite set of parametersαi_{, as the linear}

will be written as U(α), with the projective representations of another element written as U(β). Then the projective representation composition rule (Equation 2.9) becomes:

U(α)U(β) =eiφ(α,β)U(f(α, β)), (2.12) withf(α, β) some function of the parameters that satisfies the following condition:

fi(α,0) =fi(0, α) = αi, fi(0,0) = 0.

A power series expansion of this function centred at zero is given, as in Equation 2.5, by:

fi(α, β) =αi+βi+f_jki αjβk.

A Taylor series expansion of a projective representation is given by:

U(α) = In+Xiαi+

1 2Xijα

i_αj_+...

where Xi and Xij are respectively first- and second-order partial derivatives ofU

with respect to the parameters. The left-hand-side of Equation 2.12 is therefore, from Equation 2.6: LHS=In+Xiαi+Xiβi+XiαiXjβj+ 1 2Xijα i_αj ₊1 2Xijβ i_βj _+...

The Taylor series expansion of the phase centred at the identity is given by:

φ(α, β) =fijαiβj +...

wherefij is some constant. The lower order terms vanish due to the condition that

φ(α,0) = φ(0, α) = 0. Therefore the expansion of the right-hand-side of Equation 2.12 is the same as given in Equation 2.7, except for an additional term given by

fijαiβj. Then, still following the method of Section 2.1.3 and subtracting common

terms from the left-hand-side and right-hand-side in the expansion of Equation 2.12 yields:

XiXjαiβj =Xkfkijαiβj+Xijαiβj+fijαiβj

⇒ Xij =XiXj −Xkfkij +fij.

But Xij is symmetric, and so:

XiXj−Xkfkij+fij =XjXi−Xkfkji+fji

[Xi, Xj] =CkijXk+Cij, (2.13)

where Cij :=−fij +fji =−Cji. This is identical to Equation 2.8 except for the

addition of n(n₂−1) termsCij which are called thecentral charges of the Lie algebra,

Equation 2.13 must satisfy the Jacobi identity, so: 0 = [Xi,[Xj, Xk]] + [Xj,[Xk, Xi]] + [Xk,[Xi, Xj]]

=C`jk[Xi, X`] +C`ki[Xj, X`] +C`ij[Xk, X`]

=C`jk(Cmi`+Ci`In) +C`ki(Cmj`+Cj`In) +C`ij(Cmk`+Ck`In),

which leads to two conditions on the coefficients:

C`jkCmi`+C`kiCmj`+C`ijCmk` = 0 (2.14)

and

C`jkCi`+C`kiCj`+C`ijCk` = 0. (2.15)

Equation 2.14 leads to an obvious solution of Equation 2.15:

Cij =Cmijθm. (2.16)

When the coefficientsCij are replaced in this way, then the Lie algebra generators

can be redefined to eliminate the central charges of Equation 2.13, by defining new generators:

˜

Xi :=Xi+θi. (2.17)

The Lie algebra of these new generators is of the form of Equation 2.8:

h

˜

Xi,X˜j

i

=CkijX˜k. (2.18)

If Equation 2.16 is the only solution to Equation 2.15, then there are no intrinsic central charges - the central charges that appear can always be discarded by the trivial redefinition of the basis of the algebra given by Equation 2.17.

If however there exist solutions of Equation 2.15 other than that given in Equation 2.16, then there exist intrinsic central charges to the Lie algebra. In this case the Lie algebra can be extended to include new generatorsMij that have

the central charges Cij as their eigenvalues. This is called an (algebraic) central

extension of the Lie algebra. This extended Lie algebra consists of the set of generators {Xi, Mij}, i, j = 1,2, ..., n along with all linear combinations of these

generators. The defining relations of the generators are:

[Xi, Xj] =CkijXk+Mij, (2.19)

[Xi, Mij] = 0. (2.20)

If the Lie group representations U are unitary, then the corresponding central generators Mij if they exist are proportional to the identity due to Schur’s lemma

(cf. Section 2.1.4):

and

[Xi, Xj] =CkijXk+CijIn.

This extended Lie algebra has a corresponding Lie group. If G is a Lie group which corresponds to the Lie algebra of Equation 2.18, consisting of elements

{eai_X

i}_{where a}i _{is an n-tuple, and} e

G is a Lie group which corresponds to the Lie algebra of Equations 2.19 and 2.20, consisting of elements {eai_X

i_{, e}bijMij}_{, then} e

G

is called the central extension of G. This will be referred to as the “algebraic” central extension of G.

A given Lie groupG may or may not admit an algebraic central extension. If it does not admit an algebraic central extension then there might still be intrinsically projective representations of the group indicating that the group is not its own central extension. Indeed, quite apart from the algebraic central extension, there is the possibility of a topological central extension toG. This possibility is discussed in Section 2.2.2. Central extensions are important since whenever there exist intrinsic projective representations of a group, these projective representations are isomorphic to linear representations of a central extension of the group:

U(G)∼=ρ(Ge).

However, these linear representations ρ(Ge) are not necessarily matrix representa-

tions, even if G is a matrix group (cf. [37]).