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Summary: A NOTE ON STRONGLY SEPARABLE ALGEBRAS
MARCELO AGUIAR
Abstract. Let A be an algebra over a eld k. If M is an A{bimodule, we let
M A and MA denote respectively the k{spaces of invariants and coinvariants of
M , and 'M : M A ! MA be the natural map. In this note we characterize those
algebras A for which 'M is a natural isomorphism as those separable algebras
for which the separability idempotent can be chosen to be symmetric, or as those
nite dimensional algebras for which the trace form is non-degenerate. These
algebras are called strongly separable. We also prove that the cotensor product of
C{bicomodules has a right adjoint if and only if the coalgebra C is cosemisimple,
and show that if C is strongly separable then the adjoint is given by maps of
C{bicomodules.
1. Introduction
Let k be a eld, G a group such that jGj 6= 0 in k and M a G{space. Then the
natural map between the spaces of G{invariants and G{coinvariants
M G = fm 2 M = gm = m 8 g 2 Gg ,! M !! MG = M=hgm m = g 2 G; m 2 Mi
is an isomorphism, since the map
MG !M G ;
m 7! 1
jGj
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