N ov 2 00 3 Symmetric Coalgebras

We construct a structure of a ring with local units on a co-Frobenius coalgebra. We study a special class of co-Frobenius coalgebras whose objects we call symmetric coalgebras. We prove that any semiperfect coalgebra can be embedded in a symmetric coalgebra. A dual version of Brauer's equivalence theorem is presented, allowing a characterization of symmetric coalgebras by comparing certain functors. We define an automorphism of the ring with local units constructed from a co-Frobenius coalgebra, which we call the Nakayama automorphism. This is used to give a new characterization to symmetric coalgebras and to describe Hopf algebras that are symmetric as coalgebras. As a corollary we obtain as a consequence the known characterization of Hopf algebras that are symmetric as algebras. Frobenius algebras appeared in group representation theory around 100 years ago. After-wards they were recognized in many fields of mathematics: commutative algebra, topology, quantum field theory, von Neumann algebras, Hopf algebras, quantum Yang-Baxter equation , etc; see [6], [17] for classical aspects, and [5], [11], [15], [24] for more recent developments. For instance in [15] ideas about Frobenius algebras, Hopf subalgebras, solutions of the Yang-Baxter equation, the Jones polynomial and 2-dimensional topological quantum * On leave from University of Bucharest, Dept. Mathematics.