monoid bialgebra is a Hopf algebra if and only if monoid is a group

Assume that H is a Hopf algebra with comultiplication Δ, counit ε and antipode S. It is well known, that if cH and Δ(c)=i=1naibi, then i=1nS(ai)bi=ε(c)1=i=1naiS(bi) (actualy, this condition defines the antipode), where on the left and right side we have multiplication in H.

Now let G be a monoid and k a field. It is well known that kG is a bialgebraPlanetmathPlanetmathPlanetmath (please, see parent object for details), but one may ask, when kG is a Hopf algebra? We will try to answer this question.

PropositionPlanetmathPlanetmath. A monoid bialgebra kG is a Hopf algebra if and only if G is a group.

Proof. ,,” If G is a group, then define S:kGkG by S(g)=g-1. It is easy to check, that S is the antipode, thus kG is a Hopf algebra.

,,” Assume that kG is a Hopf algebra, i.e. we have the antipode S:kGkG. Then, for any gG we have S(g)g=gS(g)=1 (because Δ(g)=gg and ε(g)=1). Here 1 is the identityPlanetmathPlanetmathPlanetmath in both G and kG. Of course S(g)kG, so


Thus we have


Of course G is a basis, so this decomposition is unique. Therefore, there exists gG such that λg=1 and λh=0 for hg. We obtain, that 1=gg, thus g is left invertible. Since g was arbitrary it implies that g is invertible. Thus, we’ve shown that G is a group.

Title monoid bialgebra is a Hopf algebra if and only if monoid is a group
Canonical name MonoidBialgebraIsAHopfAlgebraIfAndOnlyIfMonoidIsAGroup
Date of creation 2013-03-22 18:58:51
Last modified on 2013-03-22 18:58:51
Owner joking (16130)
Last modified by joking (16130)
Numerical id 4
Author joking (16130)
Entry type Theorem
Classification msc 16W30