Hopf order

Let be a finite extension of the -adic rationals endowed with the -adic valuation with and let be its ring of integers (cf. Extension of a field; Norm on a field; -adic number). Let be the group ring of a finite group (cf. also Group algebra; Cross product), with . An -Hopf order in is a rank- -Hopf algebra (cf. Hopf algebra) satisfying as -Hopf algebras.

There is a method [a2] for constructing -Hopf orders in using so-called -adic order-bounded group valuations on . Given a -adic order-bounded group valuation , let be an element in of value . Then the -Hopf order in determined by (called a Larson order) is of the form

For Abelian (cf. Abelian group), the classification of -Hopf orders in is reduced to the case where is a -group. Specifically, one takes , cyclic of order , and assumes that contains a primitive th root of unity, denoted by . In this case, a -adic order-bounded group valuation on is determined by its values for , , and the Larson order is denoted by

It is known [a3] that every -Hopf order in can be written as a Tate–Oort algebra , which in turn can be expressed as the Larson order

Thus, every -Hopf order in is Larson. For this is not the case, though every -Hopf order does contain a maximal Larson order [a2].

For there exists a large class of -Hopf orders in (called Greither orders), of the form

, where and are values from a -adic order-bounded group valuation on and is an element in the Larson order (see [a1]). The parameter is an element in the units group , where is the ramification index of in , and . If , then the Greither order is the Larson order ; moreover, if and only if .

Since , the linear dual of the -Hopf order in is an -Hopf order in . One has

and

where , (see [a5]). It is known [a4] that an arbitrary -Hopf order in is either a Greither order or the linear dual of a Greither order. Thus, every -Hopf order in can be written in the form

for some , , .

The construction of Greither orders can be generalized to give a complete classification of -Hopf orders in , as well as a class of -Hopf orders in , , which are not Larson (see [a5]). However, the complete classification of -Hopf orders in , , remains an open problem.

See also Hopf orders, applications of.

References

[a1]	C. Greither, "Extensions of finite group schemes, and Hopf Galois theory over a complete discrete valuation ring" Math. Z. , 210 (1992) pp. 37–67 Zbl 0737.11038
[a2]	R.G. Larson, "Hopf algebra orders determined by group valuations" J. Algebra , 38 (1976) pp. 414–452 Zbl 0407.20007
[a3]	J. Tate, F. Oort, "Group schemes of prime order" Ann. Sci. Ecole Norm. Super. (4) , 3 (1970)
[a4]	R.G. Underwood, "$R$-Hopf algebra orders in $KC_{p^2}$" J. Algebra , 169 (1994},) pp. 418–440 Zbl 0820.16036
[a5]	R.G. Underwood, "The valuative condition and $R$-Hopf algebra orders in $KC_{p^3}$" Amer. J. Math. (4) , 118 (1996) pp. 701–743 Zbl 0857.16039
[b1]	R.G. Underwood, "An Introduction to Hopf Algebras" Springer (2011) ISBN 978-0-387-72765-3 Zbl 1234.16022