  A fundamental problem in the theory of Hopf algebras is the classification
and construction of finite-dimensional (minimal) triangular Hopf algebras (A,R)
introduced by Drinfeld. Only recently Etingof and the author completely solved
this problem for semisimple A over algebraically closed fields of
characteristics 0 and p>>dim(A) (any p if one assumes that A is also
cosemisimple). In this paper we take the first step towards solving this
problem for finite-dimensional pointed Hopf algebras over an algebraically
closed field k of characteristic 0. We first prove that the fourth power of the
antipode of any triangular pointed Hopf algebra A is the identity. We do that
by focusing on minimal triangular pointed Hopf algebras (A,R) (every triangular
Hopf algebra contains a minimal triangular sub Hopf algebra) and proving that
the group algebra of the group of grouplike elements of A (which must be
abelian) admits a minimal triangular structure and consequently that A has the
structure of a biproduct. We also generalize our result on the order of the
antipode to any finite-dimensional quasitriangular Hopf algebra A whose
Drinfeld element u acts as a scalar in any irreducible representation of A
(e.g. when A^* is pointed). Second, we describe a method of construction of
finite-dimensional pointed Hopf algebras which admit a minimal triangular
structure, and classify all their minimal triangular structures. We conclude
the paper by proving that any minimal triangular Hopf algebra which is
generated as an algebra by grouplike elements and skew primitive elements is
isomorphic to a minimal triangular Hopf algebra constructed using our method.
