44 Noncommutative Mathematics for Quantum Systems
are satisfied, then the pair(A,∆)is called acompact quantum group
(CQG).
If the algebraAof a compact quantum group is commutative,
thenAis isomorphic to the algebraC(G)of continuous functions
on a compact groupG. To emphasize that for an arbitrary (that is,
not necessarily noncommutative) compact quantum group(A,∆)
the algebraAreplaces the algebra of continuous functions on an
(abstract) quantum analog of a group, the notationG= (A,∆)and
A=C(G)is also frequently used.
The map ∆ is called the coproduct of A and it induces the
convolution productof functionals
λ?μ:= (λ⊗μ)◦∆, λ,μ∈A′.The following fact is of the fundamental importance,cf.[Wor98,
Theorem 2.3].
Proposition 1.6.2 LetAbe a compact quantum group. There exists a
unique state h∈A′(called theHaar stateofA) such that for all a∈A
(h⊗idA)◦∆(a) = (idA⊗h)◦∆(a) =h(a)1.In general, the Haar state of a compact quantum group need not
be faithful or tracial.
Corepresentations
An elementu= (ujk) 1 ≤j,k≤n ∈ Mn(A)is called ann-dimensional
corepresentation of G = (A,∆) if for all j,k = 1,... ,nwe have
∆(ujk) =∑np= 1 uj p⊗upk. All corepresentations considered in this
course are supposed to be finite-dimensional. A corepresentationu
is said to benon-degenerate, ifuis invertible,unitary, ifuis unitary,
andirreducible, if the only matricesT∈Mn(C)withTu=uTare
multiples of the identity matrix. Two corepresentationsu,v ∈
Mn(A) are called equivalent, if there exists an invertible matrix
U∈Mn(C)such thatUu=vU.
An important feature of compact quantum groups is the
existence of a dense∗-subalgebraA(the algebra of thepolynomials
of A), which is in fact a Hopf ∗-algebra – so, for example
∆:A → A⊗A. With the notationG= (A,∆), one often refers to
Aas Pol(G).
Fix a complete family (u(s))s∈I of mutually inequivalent
irreducible unitary corepresentations ofA, then{u(ks`);s∈ I, 1≤k,