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


Let H=š•„n⁢(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 ε:H→k such that ε 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 ker⁢ε=0 (because ε⁢(1)=1). ContradictionMathworldPlanetmathPlanetmath, because dimk⁢H>1=dimk⁢k.

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

Δ⁢(Ei⁢j)=āˆ‘p=1nEi⁢pāŠ—Ep⁢j;
ε⁢(Ei⁢j)=Γi⁢j,

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

Now assume that matrix coalgebra H=š•„c⁢(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 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
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