  Following the ideas of our previous works math.QA/0008232 (joint with
Andruskiewitsch) and math.QA/0101049, we study families of triangular Hopf
algebras obtained by twisting finite supergroups by a twist lying entirely in
the odd part. These families are parametrized by data (G,V,u,B), where G is a
finite group, V its finite dimensional representation, u a central element of G
of order 2 acting by -1 on V, and B an element of S^2V. We fix the discrete
data G,V,u, and find the set of isomorphism classes of the members of the
family as Hopf algebras, in terms of the continuous parameter B. This set is
often infinite, which provides examples of nontrivial continuous families of
triangular Hopf algebras. The lowest dimension in which such a family occurs is
32, in which case we get 3 families which are dual to the 3 families of pointed
Hopf algebras of dimension 32 constructed recently by Grana. Furthermore, we
show that if (S^2V)^G=0 then such continuous families are nontrivial not only
up to a Hopf algebra isomorphism, but also up to twisting of the
multiplication. Thus, they provide counterexamples to Masuoka's weakened
Kaplansky's 10th conjecture, which claims that up to twisting, there are
finitely many types of Hopf algebras in each dimension. Finally, we study the
algebra structure of the duals of our families, and show they are direct sums
of Clifford algebras. Since there are finitely many types of Clifford algebras
in each dimension, this allows us to construct nontrivial families of rigid
tensor structures on the abelian category of modules over a finite dimensional
algebra, with a fixed Grothendieck ring.
