Frobenius monads and pseudomonoids Introduction - ResearchGate

algebra is a quasibialgebra H together with an algebra anti-morphism S: H æÆ æ H (called
the antipode) and a bimodule isomorphism p :M@ N.
This is equivalent to the original
definition of [Dd] that, instead of the isomorphism p , involved two elements a
H satisfying the equations
and b of
Ec () a = ÂSc ( i) aci
1 2 , Ec () b = Âci bSc
( i )
1 2 1 2 3
, Â fi bS( fi ) afi
= 1
-1 -2 -3
and  S( fi ) a fi bS( fi
) = 1
i
i
i
(see Section 2.4 of [Maj]). We shall say a bit more about the equivalence of the definitions
soon.
Take V = Vect k and recall that a k-algebra A is a one-object V-category SA. It is
obvious that an algebra morphism f: A æÆ æ B is the same as a V-functor f: SA æÆ æ SB.
So what is a V-natural transformation s :ffi g in terms of the algebra morphisms f and
g from A to B ? It is nothing other than an element a of B such that afa ( ) = ga ( ) a for
all elements a of A. Therefore we see that f belongs in the square
in
SH
D
SH ƒ SH
fl f
D Dƒ1 SH ƒ SH
SH ƒ SH ƒ SH
1 ƒD
V - Cat. From this it follows easily that: a quasibialgebra structure on an algebra H is
precisely a normalized pseudomonoid structure on SH in
V - Cat coop
( ) .
Now we move from V - Cat coop
( ) to V -Mod coop
( ) where V-functors have right
adjoints and the object SH op is a bidual for SH. Here we can observe that the right adjoint
*
D : SH ƒ SH æ æÆ
SH
of D is the bimodule M above and the composite module
SH ƒ D SHƒS ƒ SH op e ƒ SH
SH ƒ SH æææÆ SH ƒ SH ƒ SH æææææÆ SH ƒ SH ƒ SH ææ æÆ
SH
is N. It follows that: a quasi-Hopf structure on the quasialgebra H is precisely a left
autonomous structure on the normalized pseudomonoid SH in
Therefore Proposition 3.1 applies to yield a Frobenius structure on SH in
i
coop
( V -Mod ) .
coop
( V -Mod )
using E * . This means that SH becomes a pseudomonoid in V -Mod using E * and D * of course, this is not the original algebra structure on H.
;
Finally in this example, we can say something about the equivalent definitions of
quasi-Hopf algebra: specifically about how p is obtained from a and f . The abovementioned
equation satisfied by the element a of H says precisely that it is a (bi)module
morphism from E to eo ( 1 ƒ S)
o D (note that E is k and eo ( 1 ƒ S)
o D is H with
appropriate actions). Then p is the pasting composite of the following diagram.
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