Given an algebra C over a
commutative ring k and an element (called a C-two-cocycle) σ =∑iai⊗ bi⊗ ci in
C ⊗kC ⊗kC satisfying certain relations, Sweedler defined a new multiplication ∗ on
C by x∗y =∑iaixbiyci for all x, y in C and denoted C with this new multiplication
by Cσ. This paper studies three rigidity properties which arise by asking
whether:
Cσ≃ C as algebras;
a certain functor from the category of C-bimodules to the category of
Cσ-bimodules is an equivalence;
a certain functor from the category of algebras over C to the category of
algebras over Cσ is an equivalence.
For certain algebras over a field k (including finite dimensional algebras possessing a
Wedderburn factor), these rigidity properties are shown to be equivalent to
(respectively): (i) all k-separable subalgebras B of C are commutative and for a
separability idempotent ∑ixi⊗yi of B,{c ∈ C|∑ixicyi= 0} is an ideal with square
{0}; (ii) all k-separable subalgebras of C are central; (iii) k is the only k-separable
subalgebra of C.
