example of algebras and coalgebras which cannot be turned into Hopf algebras
Let be a matrix algebra over a field with standard multiplication and assume that . Assume that can be turned into a Hopf algebra. In particular, there is such that is a morphism of algebras. It can be shown that matrix algebra is simple, i.e. if is a two-sided ideal, then or . Thus we have that (because ). Contradiction, because .
Now consider a vector space of all matrices over . We introduce coalgebra structure on . Let be a matrix in with in place and everywhere else. Of course forms a basis of and it is sufficient to define comultiplication and counit on it. Define
Now assume that matrix coalgebra (where ) can be turned into a Hopf algebra. Since is finite dimensional, then we can take dual Hopf algebra . But the underlaying algebra structure of is isomorphic to a matrix algebra (as we remarked earlier), which we’ve already shown to be impossible. Thus matrix coalgebra cannot be turned into a Hopf algebra.
|Title||example of algebras and coalgebras which cannot be turned into Hopf algebras|
|Date of creation||2013-03-22 18:58:45|
|Last modified on||2013-03-22 18:58:45|
|Last modified by||joking (16130)|