Noncommutative Mathematics for Quantum Systems

Independence and L ́evy Processes in Quantum Probability 99

can be understood as a functor fromComAlgto the category of
unital semigroups. The multiplication in MorAlg(B,A)is given by
the convolution, that is,

f?g=mA◦(f⊗g)◦∆B

and the unit element isεB (^1) A : B 3b 7→εB(b) (^1) A ∈ A. A unit-
preserving algebra homomorphismh:A 1 → A 2 gets mapped to
the unit-preserving semigroup homomorphism
MorComAlg(B,A 1 ) 3 f→h◦f∈MorComAlg(B,A 2 ),
since
h◦(εB (^1) A 1 ) = εB (^1) A 2 ,
h◦(f?g) = (h◦f)?(h◦g),
forA 1 ,A 2 ∈ ObComAlg,h ∈ MorComAlg(A 1 ,A 2 ), and algebra
homomorphismsf,g∈MorComAlg(B,A 1 ).
IfBis a commutative Hopf algebra with antipodeS, then the
functor MorComAlg(B,·)takes values in the category of groups, since
then MorComAlg(B,A)is a group with respect to the convolution
product. The inverse of a homomorphismf:B → Awith respect
to the convolution product is given byf◦S.
The calculation
(f?g)(ab) = mA◦(f⊗g)◦∆B(ab)
= f(a( 1 )b( 1 ))g(a( 2 )b( 2 )) =f(a( 1 ))f(b( 1 ))g(a( 2 ))g(b( 2 ))
= f(a( 1 ))g(a( 2 ))f(b( 1 ))g(b( 2 )) = (f?g)(a)(f?g)(b)
shows that the convolution productf?gof two homomorphisms
f,g:B → Ais again a homomorphism. It also gives an indication
why noncommutative bialgebras or Hopf algebras do not give rise
to a similar functor on the category of noncommutative algebras,
since we had to commutef(b( 1 ))withg(a( 2 )).
Zhang [Zha91], Bergman, and Hausknecht [BH96] showed that
if one replaces the tensor product in the definition of bialgebras
and Hopf algebras by the free product, then one arrives at a class
of algebras that do give rise to a functor from the category of
noncommutative algebras to the category of semigroups or
groups.