Noncommutative Mathematics for Quantum Systems

Independence and L ́evy Processes in Quantum Probability 45

`≤ns}(wherensdenotes the dimension ofu(s)) is a linear basis of
Pol(G),cf.[Wor98, Proposition 5.1]. We shall reserve the index
s=0 for the trivial corepresentationu(^0 ) = 1. The Hopf algebra
structure on Pol(G)is defined by

ε(u(jks)) =δjk, S(u(jks)) = (u(kjs))∗ forj,k=1,... ,ns,

whereε: Pol(G)→Cis the counit andS: Pol(G)→Pol(G)is the
antipode. They satisfy

(id⊗ε)◦∆ = id= (ε⊗id)◦∆, (1.6.1)

mA◦(id⊗S)◦∆ = ε(a) 1 =mA◦(S⊗id)◦∆, (1.6.2)

(

S(a∗)∗

)

= a (1.6.3)

for alla∈Pol(G). Let us also remind that the Haar state is always
faithful on Pol(G).

SetVs = span{ujk(s); 1 ≤ j,k ≤ ns}fors ∈ I. Letu(s)be an

irreducible unitary corepresentation of G. Then u(s) =
(
u(jks)

∗)
1 ≤j,k≤ns is also an irreducible non-degenerate
corepresentation, but no longer unitary. By [Wor98, Proposition

5.2], there exists an irreducible unitary corepresentation u(s

c)
,
called thecontragredientcorepresentation ofu(s), that is equivalent

tou(s). We haveVs∗=Vscand(sc)c=s.

The dual discrete quantum group
To every compact quantum groupG= (A,∆)there exists a dual
discrete quantum groupGˆ,cf.[PW90]. For our purposes it will be
most convenient to introduce Gˆ in the setting of Van Daele’s
algebraic quantum groups,cf.[VD98, VD03]. However, the reader
should be aware that we adopt a slightly different convention for
the Fourier transform.
A pair(A,∆), consisting of a ∗-algebra A(with or without
identity) and a coassociative comultiplication∆:A→M(A⊗A),
is called an algebraic quantum group if the product is
non-degenerate (that is,ab=0 for allaimpliesb=0), if the two
operatorsT 1 : A⊗A 3 a⊗b 7→ ∆(a)(b⊗ 1 ) ∈ A⊗Aand
T 2 : A⊗A 3 a⊗b 7→ ∆(a)( 1 ⊗b) ∈ A⊗Aare well defined
bijections and if there exists a non-zero left invariant positive