grouplike elements in Hopf algebras
Now let be a Hopf algebra over a field , with identity , comultiplication , counit and antipode . Recall that element is called grouplike iff and . The set of all grouplike elements is nonempty, because . Also, since comultiplication is an algebra morphism, then is multiplicative, i.e. if , then . Furthermore, it can be shown that for any we have and . Thus is a group under multiplication inherited from .
It is easy to see, that the vector subspace spanned by is a Hopf subalgebra of isomorphic to . It can be shown that is always linearly independent, so if is finite dimensional, then is a finite group. Also, if is finite dimensional, then it follows from the Nichols-Zoeller Theorem, that the order of divides .
From these observations it follows that if is a prime number, then is either trivial or the order of is equal to (i.e. is cyclic of order ). The second case implies that is isomorphic to and it can be shown that the first case cannot occur.
|Title||grouplike elements in Hopf algebras|
|Date of creation||2013-03-22 18:58:39|
|Last modified on||2013-03-22 18:58:39|
|Last modified by||joking (16130)|