nichols-zoeller theorem

Let H be a Hopf algebra over a field k with an antipode S. We will say that KH is a Hopf subalgebraPlanetmathPlanetmathPlanetmath if K is both subalgebra and subcoalgebra of underlaying algebra and coalgebra structuresMathworldPlanetmath of H, and additionaly S(K)K. In particular a Hopf subalgebra KH is an algebra over k, so H may be regarded as a K-module.

The Nichols-Zoeller Theorem. If KH 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 similarPlanetmathPlanetmath 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 n2 and let TH be the upper triangular matrixMathworldPlanetmath 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 n2. 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