Abstract
Let H be a Hopf algebra over a field K and assume a K-algebra A is an H-module algebra under two actions; · and ∘. We call these actions cocycle equivalent if there is an action of H on M
2(A), h ♦ X, such that
for
h ∈
H,
X ∈
M
2(
A) and
a,
b ∈
A. Two actions are cocycle equivalent if and only if there are cocycles that relate the two actions. Using these, it is shown that cocycle equivalence is a equivalence relation. Finally let
H be a finite dimensional, semisimple, cocommutative Hopf algebra and assume
K is a splitting field of
H. It is shown that the Connes spectrum of
H acting on
M
2(
A) is the intersection of the Connes spectra of
H acting on
A under · and ∘. Denote the smash product of
A and
H under the action · by (
A#
H,·). Let
A be
H-prime, then (
A#
H,·) is prime if and only if (
A)#H, ♦) is prime if and only if (
M
2(
A)#
H, ♦) is prime.
ACKNOWLEDGMENT
The author's research was supported by the Charles Phelps Taft Memorial Fund.