The Adams operators
on a Hopf algebra
are the convolution powers of the identity of
.
They are also called Hopf powers or Sweedler powers. We study the Adams operators
when
is graded connected. The main result is a complete description of the characteristic
polynomial — both eigenvalues and their multiplicities — for the action of the operator
on each homogeneous
component of
. The eigenvalues
are powers of
. The multiplicities
are independent of
,
and in fact only depend on the dimension sequence of
.
These results apply in particular to the antipode of
, as
the case
.
We obtain closed forms for the generating function of the sequence of traces
of the Adams operators. In the case of the antipode, the generating function bears
a particularly simple relationship to the one for the dimension sequence. In the case
where
is cofree, we give an alternative description for the characteristic polynomial
and the trace of the antipode in terms of certain palindromic words.
We discuss parallel results that hold for Hopf monoids in species and for
-Hopf algebras.
