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:Gk|f(g)=0 for almost all gG}.

We identify gG with a function fg:Gk such that fg(g)=1 and fg(h)=0 for hg. Thus, every element in kG is of the form


for λgk. The vector space kG can be turned into a k-algebra, if we define multiplication as follows:


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 elementPlanetmathPlanetmath of G is the identityPlanetmathPlanetmathPlanetmath in kG.

Furthermore, we can turn kG into a coalgebra as follows: comultiplication Δ:kGkGkG is defined by Δ(g)=gg and counit ε:kGk is defined by ε(g)=1. One can easily check that this defines coalgebra structure on kG.

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

Title monoid bialgebra
Canonical name MonoidBialgebra
Date of creation 2013-03-22 18:58:48
Last modified on 2013-03-22 18:58:48
Owner joking (16130)
Last modified by joking (16130)
Numerical id 4
Author joking (16130)
Entry type Example
Classification msc 16W30