nichols-zoeller theorem
Let H be a Hopf algebra over a field k with an antipode S. We will say that K⊆H is a Hopf subalgebra if K is both subalgebra and subcoalgebra of underlaying algebra and coalgebra structures
of H, and additionaly S(K)⊆K. In particular a Hopf subalgebra K⊆H is an algebra over k, so H may be regarded as a K-module.
The Nichols-Zoeller Theorem. If K⊆H is a Hopf subalgebra of a Hopf algebra H, then H is free as a K-module. In particular, if H is finite dimensional, then dimkK divides dimkH.
Remark 1. This theorem shows that Hopf algebras are very similar to groups, because this is a Hopf analogue of the Lagrange Theorem.
Remark 2. Generally this theorem does not need to hold if H is only an algebra. For example, consider H=𝕄n(k) the matrix algebra, where n≥2 and let T⊆H be the upper triangular matrix subalgebra. It is well known that dimkH=n2 and dimkT=n(n+1)2. Of course n(n+1)2 does not divide n2 for n≥2. Thus the Nichols-Zoeller Theorem does not hold for algebras.
Title | nichols-zoeller theorem |
---|---|
Canonical name | NicholszoellerTheorem |
Date of creation | 2013-03-22 18:58:34 |
Last modified on | 2013-03-22 18:58:34 |
Owner | joking (16130) |
Last modified by | joking (16130) |
Numerical id | 6 |
Author | joking (16130) |
Entry type | Theorem |
Classification | msc 16W30 |