Editing Projective representation (section)

===Finite-dimensional projective unitary representations===
In quantum physics, [[Symmetry in quantum mechanics|symmetry]] of a physical system is typically implemented by means of a projective unitary representation <math>\rho</math> of a Lie group <math>G</math> on the quantum [[Hilbert space]], that is, a continuous homomorphism
:<math>\rho: G\rightarrow\mathrm{PU}(\mathcal H),</math>
where <math>\mathrm{PU}(\mathcal H)</math> is the quotient of the unitary group <math>\mathrm{U}(\mathcal H)</math> by the operators of the form <math>cI,\,|c| = 1</math>. 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 [[Complex projective space|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 <math>G</math> then gives rise to a projective unitary representation <math>\rho_*</math> of the Lie algebra <math>\mathfrak g</math> of <math>G</math>. In the finite-dimensional case, it is always possible to "de-projectivize" the Lie-algebra representation <math>\rho_*</math> simply by choosing a representative for each <math>\rho_*(X)</math> having trace zero.<ref>{{harvnb|Hall|2013}} Proposition 16.46</ref> In light of the [[Lie group–Lie algebra correspondence#The correspondence|homomorphisms theorem]], it is then possible to de-projectivize <math>\rho</math> itself, but at the expense of passing to the universal cover <math>\tilde G</math> of <math>G</math>.<ref>{{harvnb|Hall|2013}} Theorem 16.47</ref> That is to say, every finite-dimensional projective unitary representation of <math>G</math> arises from an ordinary unitary representation of <math>\tilde G</math> 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 <math>G</math> arises from a ''determinant-one'' ordinary unitary representation of <math>\tilde G</math> (i.e., one in which each element of <math>\tilde G</math> acts as an operator with determinant one). If <math>\mathfrak g</math> is semisimple, then every element of <math>\mathfrak g</math> is a linear combination of commutators, in which case ''every'' representation of <math>\mathfrak g</math> is by operators with trace zero. In the semisimple case, then, the associated linear representation of <math>\tilde G</math> is unique.

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

An important example is the case of [[Rotation group SO(3)|SO(3)]], whose universal cover is [[Rotation group SO(3)#Connection between SO(3) and SU(2)|SU(2)]]. Now, the Lie algebra <math>\mathrm{su}(2)</math> 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.<ref>{{harvnb|Hall|2015}} proof of Theorem 4.28</ref> Thus, the irreducible ''projective'' representations of SO(3) are in one-to-one correspondence with the irreducible ''ordinary'' representations of SU(2).