Double Constructions of Frobenius Algebras and Nondegenerate Connes 2-cocycles and Their Duality

We construct an associative algebra with a decomposition into a direct sum of the underlying vector spaces of another associative algebra and its dual space such that both of them are subalgebras and the natural symmetric bilinear form is invariant or the natural skew-symmetric bilinear form is a Connes 2-cocycle. Both of them are equivalent to a kind of bialgebra, namely, associative bialgebra and dendriform bialgebra respectively. By comparing associative bialgebras and dendriform bialgebras, we observe that there is a clear analogy between them. Due to the correspondences between certain symmetries and skew-symmetries appearing in the analogy, we regard it as a kind of duality.