Difference between revisions of "Hopf order"
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<TR><TD valign="top">[a4]</TD> <TD valign="top"> R.G. Underwood, "$R$-Hopf algebra orders in $KC_{p^2}$" ''J. Algebra'' , '''169''' (1994},) pp. 418–440 {{ZBL|0820.16036}}</TD></TR> | <TR><TD valign="top">[a4]</TD> <TD valign="top"> R.G. Underwood, "$R$-Hopf algebra orders in $KC_{p^2}$" ''J. Algebra'' , '''169''' (1994},) pp. 418–440 {{ZBL|0820.16036}}</TD></TR> | ||
<TR><TD valign="top">[a5]</TD> <TD valign="top"> 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}}</TD></TR> | <TR><TD valign="top">[a5]</TD> <TD valign="top"> 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}}</TD></TR> | ||
+ | |||
+ | <TR><TD valign="top">[b1]</TD> <TD valign="top"> R.G. Underwood, "An Introduction to Hopf Algebras" Springer (2011) ISBN 978-0-387-72765-3 {{ZBL|1234.16022}}</TD></TR> | ||
</table> | </table> |
Revision as of 19:59, 12 January 2018
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 |
Hopf order. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Hopf_order&oldid=42726