example of algebras and coalgebras which cannot be turned into Hopf algebras

Let H=š•„nā¢(k) be a matrix algebra over a field k with standard multiplicationPlanetmathPlanetmath and assume that n>1. Assume that H can be turned into a Hopf algebra. In particular, there is Īµ:Hā†’k such that Īµ is a morphism of algebrasPlanetmathPlanetmath. It can be shown that matrix algebra is simple, i.e. if IāŠ†H is a two-sided idealMathworldPlanetmath, then I=0 or I=H. Thus we have that kerā¢Īµ=0 (because Īµā¢(1)=1). ContradictionMathworldPlanetmathPlanetmath, because dimkā¢H>1=dimkā¢k.

Now consider H=š•„cā¢(n,k) a vector spaceMathworldPlanetmath of all nƗn matrices over k. We introduce coalgebra structureMathworldPlanetmath on H. Let Eiā¢j be a matrix in H with 1 in (i,j) place and 0 everywhere else. Of course {Eiā¢j} forms a basis of H and it is sufficient to define comultiplication and counit on it. Define


where Ī“iā¢j denotes Kronecker delta. It can be easily checked, that (š•„cā¢(n,k),Ī”,Īµ) is a coalgebra known as the matrix coalgebra. Also, is well known that the dual algebra š•„cā¢(n,k)* is isomorphicPlanetmathPlanetmathPlanetmath to the standard matrix algebra.

Now assume that matrix coalgebra H=š•„cā¢(n,k) (where n>1) can be turned into a Hopf algebra. Since H is finite dimensional, then we can take dual Hopf algebra H*. But the underlaying algebra structure of H* is isomorphic to a matrix algebra (as we remarked earlier), which weā€™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
Canonical name ExampleOfAlgebrasAndCoalgebrasWhichCannotBeTurnedIntoHopfAlgebras
Date of creation 2013-03-22 18:58:45
Last modified on 2013-03-22 18:58:45
Owner joking (16130)
Last modified by joking (16130)
Numerical id 4
Author joking (16130)
Entry type Example
Classification msc 16W30