Let be a Hopf algebra over a field with an antipode . We will say that is a Hopf subalgebra if is both subalgebra and subcoalgebra of underlaying algebra and coalgebra structures of , and additionaly . In particular a Hopf subalgebra is an algebra over , so may be regarded as a -module.
Remark 2. Generally this theorem does not need to hold if is only an algebra. For example, consider the matrix algebra, where and let be the upper triangular matrix subalgebra. It is well known that and . Of course does not divide for . Thus the Nichols-Zoeller Theorem does not hold for algebras.
|Date of creation||2013-03-22 18:58:34|
|Last modified on||2013-03-22 18:58:34|
|Last modified by||joking (16130)|