# monoid bialgebra

Let $G$ be a monoid and $k$ a field. Consider the vector space $kG$ over $k$ with basis $G$. More precisely,

 $kG=\{f:G\to k\ |\ f(g)=0\mbox{ for almost all }g\in G\}.$

We identify $g\in G$ with a function $f_{g}:G\to k$ such that $f_{g}(g)=1$ and $f_{g}(h)=0$ for $h\neq g$. Thus, every element in $kG$ is of the form

 $\sum_{g\in G}\lambda_{g}g,$

for $\lambda_{g}\in k$. The vector space $kG$ can be turned into a $k$-algebra, if we define multiplication as follows:

 $g\cdot h=gh,$

where on the right side we have a multiplication in the monoid $G$. This definition extends linearly to entire $kG$ and defines an algebra structure on $kG$, where neutral element of $G$ is the identity in $kG$.

Furthermore, we can turn $kG$ into a coalgebra as follows: comultiplication $\Delta:kG\to kG\otimes kG$ is defined by $\Delta(g)=g\otimes g$ and counit $\varepsilon:kG\to k$ is defined by $\varepsilon(g)=1$. One can easily check that this defines coalgebra structure on $kG$.

The vector space $kG$ is a bialgebra with with these algebra and coalgebra structures and it is called a monoid bialgebra.

Title monoid bialgebra MonoidBialgebra 2013-03-22 18:58:48 2013-03-22 18:58:48 joking (16130) joking (16130) 4 joking (16130) Example msc 16W30