100 Noncommutative Mathematics for Quantum Systems
Adual group[Voi87, Voi90] (calledH-algebraorcogroupin the
category of unital associative∗-algebras in [Zha91] and [BH96],
resp.) is a unital∗-algebraBequipped with three unital∗-algebra
homomorphisms∆:B → B‰B,S:B → B, andε:B →C(also
calledcomultiplication or coproduct,antipode, andcounit) such that
(
∆‰id
)
◦∆ =(
id‰∆)
◦∆, (1.9.1)
(
ε‰id)
◦∆= id =(
id‰ε)
◦∆, (1.9.2)
mB◦(
S‰id)
◦∆= id =mB◦(
id‰S)
◦∆, (1.9.3)wheremB :B‰B → B,mB(a 1 ⊗a 2 ⊗···⊗an) = a 1 ·a 2 · ··· ·
an, is the multiplication ofB. Besides the formal similarity, there
are many relations between dual groups on the one side and Hopf
algebras and bialgebras on the other side,cf.[Zha91]. For example,
letBbe a dual group with comultiplication∆, and letR:B‰B →
B⊗Bbe the unique unital∗-algebra homomorphism with
RB,B◦i 1 (b) =b⊗ 1 , RB,B◦i 2 (b) = 1 ⊗b,for allb ∈ B. Herei 1 ,i 2 : B → B‰B denote the canonical
inclusions ofBinto the first and the second factors of the free
product B‰B, respectively. Then B is a bialgebra with the
comultiplication∆ = RB,B◦∆, see [Zha91, Theorem 4.2], but in
general it is not a Hopf algebra.
We will not really work with dual groups, but with the
following weaker notion. Adual semigroupis a unital∗-algebraB
equipped with two unital∗-algebra homomorphisms∆ : B →
B‰Bandε :B → Csuch that Equations (1.9.1) and (1.9.2) are
satisfied. The antipode is not used in the proof of [Zha91, Theorem
4.2], and therefore we also get an involutive bialgebra(B,∆,ε)for
every dual semigroup(B,∆,ε).
Note that we can always write a dual semigroupBas a direct
sumB=C 1 ⊕B^0 , whereB^0 =kerεis even a∗-ideal. Therefore, it
is in the range of the unitization functor and the boolean,
monotone, and anti-monotone product can be defined for unital
linear functionals onB,cf.Exercise 1.8.14.
The comultiplication of a dual semigroup can also be used to
define a convolution product. Theconvolution j 1 ?j 2 of two unital
∗-algebra homomorphismsj 1 ,j 2 :B →Ais defined as
j 1 ?j 2 =mA◦(
j 1 ‰j 2)
◦∆.