grouplike elements in Hopf algebras

Recall, that if k is a field and G is a group, then the group algebra kG can be turned into a Hopf algebra, by defining comultiplication Δ(g)=gg, counit ε(g)=1 and antipode S(g)=g-1.

Now let H be a Hopf algebra over a field k, with identityPlanetmathPlanetmathPlanetmath 1, comultiplication Δ, counit ε and antipode S. Recall that element gH is called grouplike iff g0 and Δ(g)=gg. The set of all grouplike elements G(H) is nonempty, because 1G(H). Also, since comultiplication is an algebraMathworldPlanetmathPlanetmathPlanetmath morphism, then G(H) is multiplicative, i.e. if g,hG(H), then ghG(H). Furthermore, it can be shown that for any gG(H) we have S(g)G(H) and S(g)g=gS(g)=1. Thus G(H) is a group under multiplication inherited from H.

It is easy to see, that the vector subspace spanned by G(H) is a Hopf subalgebraPlanetmathPlanetmathPlanetmath of H isomorphicPlanetmathPlanetmathPlanetmath to kG(H). It can be shown that G(H) is always linearly independentMathworldPlanetmath, so if H is finite dimensional, then G(H) is a finite groupMathworldPlanetmath. Also, if H is finite dimensional, then it follows from the Nichols-Zoeller Theorem, that the order of G(H) divides dimkH.

From these observations it follows that if dimkH=p is a prime numberMathworldPlanetmath, then G(H) is either trivial or the order of G(H) is equal to p (i.e. G(H) is cyclic of order p). The second case implies that H is isomorphic to kp and it can be shown that the first case cannot occur.

Title grouplike elements in Hopf algebras
Canonical name GrouplikeElementsInHopfAlgebras
Date of creation 2013-03-22 18:58:39
Last modified on 2013-03-22 18:58:39
Owner joking (16130)
Last modified by joking (16130)
Numerical id 5
Author joking (16130)
Entry type Definition
Classification msc 16W30