# Journal of Noncommutative Geometry

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**Volume 4, Issue 4, 2010, pp. 475–530**

**DOI: 10.4171/JNCG/64**

Published online: 2010-09-01

Double constructions of Frobenius algebras, Connes cocycles and their duality

Chengming Bai^{[1]}(1) Chern Institute of Mathematics, Nankai University, China

We construct an associative algebra with a decomposition into the
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
antisymmetric bilinear form is a Connes cocycle. The former is
called a double construction of a Frobenius algebra and the latter is
called a double construction of the Connes cocycle, which is interpreted
in terms of dendriform algebras. Both of them are equivalent to a
kind of bialgebras, namely, antisymmetric infinitesimal bialgebras
and dendriform D-bialgebras, respectively. In the coboundary cases,
our study leads to what we call associative Yang–Baxter equation in
an associative algebra and `D`-equation in a dendriform algebra,
respectively, which are analogues of the classical Yang–Baxter
equation in a Lie algebra. We show that an antisymmetric solution of the
associative Yang–Baxter equation corresponds to the antisymmetric
part of a certain operator called ** O**-operator which
gives a double construction of a Frobenius algebra, whereas a
symmetric solution of the

`D`-equation corresponds to the symmetric part of an

**-operator which gives a double construction of the Connes cocycle. By comparing antisymmetric infinitesimal bialgebras and dendriform D-bialgebras, we observe that there is a clear analogy between them. Due to the correspondences between certain symmetries and antisymmetries appearing in this analogy, we regard it as a kind of duality.**

`O`
*Keywords: *Associative algebra, Frobenius algebra, Connes cocycle, Yang–Baxter equation

Bai Chengming: Double constructions of Frobenius algebras, Connes cocycles and their duality. *J. Noncommut. Geom.* 4 (2010), 475-530. doi: 10.4171/JNCG/64