Hopf order

Let $K$ be a finite extension of the $p$-adic rationals $\Q_p$ endowed with the $p$-adic valuation $\nu$ with $\nu(p)=1$ and let $R$ be its ring of integers (cf. Extension of a field; Norm on a field; $p$-adic number). Let $KG$ be the group ring of a finite group $G$ (cf. also Group algebra; Cross product), with $|G|=q$. An $R$-Hopf order in $KG$ is a rank-$q$ $R$-Hopf algebra $H$ (cf. Hopf algebra) satisfying $H\otimes_R K \cong KG$ as $K$-Hopf algebras.

There is a method [a2] for constructing $R$-Hopf orders in $KG$ using so-called $p$-adic order-bounded group valuations on $G$. Given a $p$-adic order-bounded group valuation $\xi$, let $x_{\xi(g)}$ be an element in $R$ of value $\xi(g)$. Then the $R$-Hopf order in $KG$ determined by $\xi$ (called a Larson order) is of the form

$$ A(\xi) = R \left[ \left\{ \frac{g-1}{x_{\xi(g)}} : g \in G,\ g \ne 1 \right\}\right]. $$

For $G$ Abelian (cf. Abelian group), the classification of $R$-Hopf orders in $KG$ is reduced to the case where $G$ is a $p$-group. Specifically, one takes $G = C_{p^n}$, $C_{p^n}$ cyclic of order $p^n$, and assumes that $K$ contains a primitive $p^n$th root of unity, denoted by $\zeta_n$. In this case, a $p$-adic order-bounded group valuation $\xi$ on $C_{p^n}$ is determined by its values $\xi(g^{p^i}) = s_i$ for $i=0,\ldots,n-1$, $\langle g \rangle = C_{p^n}$, and the Larson order $A(\xi)$ is denoted by

$$ H(s_{n-1}, \ldots, s_0) = R \left[ \frac{g^{p^{n-1}}-1}{x_{s_{n-1}}}, \ldots, \frac{g-1}{x_{s_0}} \right] $$

It is known [a3] that every $R$-Hopf order $H$ in $KC_p$ can be written as a Tate–Oort algebra $H_b = R[x]/\langle x^p-bx\rangle$, which in turn can be expressed as the Larson order

$$ H(s) = R \left[ \frac{g-1}{x_s} \right]. $$

Thus, every $R$-Hopf order in $KC_p$ is Larson. For $n>1$ this is not the case, though every $R$-Hopf order does contain a maximal Larson order [a2].

For $n=2$ there exists a large class of $R$-Hopf orders in $KC_{p^2}$ (called Greither orders), of the form

$$ H_v(s,r) = R \left[ \frac{g^p-1}{x_s}, \frac{g-a_v}{x_r} \right], $$

$\langle g \rangle = C_{p^2}$, where $s$ and $r$ are values from a $p$-adic order-bounded group valuation on $C_{p^2}$ and $a_v$ is an element in the Larson order $H(s)$ (see [a1]). The parameter $v$ is an element in the units group $U_{s'e + \langle re/p \rangle} \cap U_{\langle s'e /p \rangle + re}$, where $e$ is the ramification index of $p$ in $R$, and $s' = 1/(p-1)-s$. If $v=1$, then the Greither order $H_1(s,r)$ is the Larson order $H(s,r)$; moreover, $H_v(s,r) \cong H_w(s,r)$ if and only if $v/w \in U_{s'e+re}$.

Since $\zeta_n \in K$, the linear dual $H^*$ of the $R$-Hopf order $H$ in $KC_{p^n}$ is an $R$-Hopf order in $KC_{p^n}$. One has

$$ H(s)^* \cong R \left[ \frac{g-1}{x_{s'}} \right] $$

and

$$ H_v(s,r)^* \cong R \left[ \frac{g^p-1}{x_{r'}} , \frac{g-a_{v'}}{x_{s'}} \right], $$

where $v'=1+\zeta_2 - v$, $r'=1/(p-1)-r$ (see [a5]). It is known [a4] that an arbitrary $R$-Hopf order in $KC_{p^2}$ is either a Greither order or the linear dual of a Greither order. Thus, every $R$-Hopf order in $KC_{p^2}$ can be written in the form

$$ A_w(b,a) = R \left[ \frac{g^p-1}{x_b}, \frac{g-a_w}{x_a} \right] $$

for some $a,b,w$.

The construction of Greither orders can be generalized to give a complete classification of $R$-Hopf orders in $KC_{p^3}$, as well as a class of $R$-Hopf orders in $KC_{p^n}$, $n>2$, which are not Larson (see [a5]). However, the complete classification of $R$-Hopf orders in $KC_{p^n}$, $n>3$, 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