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

Assume that $H$ is a Hopf algebra with comultiplication $\mathrm{\Delta}$, counit $\epsilon $ and antipode $S$. It is well known, that if $c\in H$ and $\mathrm{\Delta}(c)=\sum _{i=1}^{n}{a}_{i}\otimes {b}_{i}$, then $\sum _{i=1}^{n}S({a}_{i}){b}_{i}=\epsilon (c)1=\sum _{i=1}^{n}{a}_{i}S({b}_{i})$ (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 bialgebra^{} (please, see parent object for details), but one may ask, when $kG$ is a Hopf algebra? We will try to answer this question.

Proposition^{}. A monoid bialgebra $kG$ is a Hopf algebra if and only if $G$ is a group.

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

,,$\Rightarrow $” Assume that $kG$ is a Hopf algebra, i.e. we have the antipode $S:kG\to kG$. Then, for any $g\in G$ we have $S(g)g=gS(g)=1$ (because $\mathrm{\Delta}(g)=g\otimes g$ and $\epsilon (g)=1$). Here $1$ is the identity^{} in both $G$ and $kG$. Of course $S(g)\in kG$, so

$$S(g)=\sum _{h\in G}{\lambda}_{h}h.$$ |

Thus we have

$$1=\left(\sum _{h\in G}{\lambda}_{h}h\right)g=\sum _{h\in G}{\lambda}_{h}hg.$$ |

Of course $G$ is a basis, so this decomposition is unique. Therefore, there exists ${g}^{\prime}\in G$ such that ${\lambda}_{{g}^{\prime}}=1$ and ${\lambda}_{{h}^{\prime}}=0$ for ${h}^{\prime}\ne {g}^{\prime}$. We obtain, that $1={g}^{\prime}g$, thus $g$ is left invertible. Since $g$ was arbitrary it implies that $g$ is invertible. Thus, we’ve shown that $G$ is a group. $\mathrm{\square}$

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