Double constructions of Frobenius algebras, Connes cocycles and their duality

Abstract

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 Frobenius algebra and the latter is called a double construction of 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 associative Yang-Baxter equation corresponds to the antisymmetric part of a certain operator called $\mathcal{O}$-operator which gives a double construction of Frobenius algebra, whereas a symmetric solution of $D$-equation corresponds to the symmetric part of an $\mathcal{O}$-operator which gives a double construction of 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 the analogy, we regard it as a kind of duality.

50 pages, 2 tables; Some straightforward proofs are omitted, some terminologies and references are changed

References in corpus (6)

• Loday-type Algebras and the Rota-Baxter Relation
• Left-symmetric Bialgebras and An Analogue of the Classical Yang-Baxter Equation
• New identities in dendriform algebras
• Infinitesimal bialgebras, pre-Lie and dendriform algebras
• O-operators on associative algebras and dendriform algebras
• $O$-operators on associative algebras and associative Yang-Baxter equations

Cited by in corpus (23)

• Leibniz bialgebras, relative Rota-Baxter operators and the classical Leibniz Yang-Baxter equation
• Infinite-dimensional Lie bialgebras via affinization of Novikov bialgebras and Koszul duality
• A bialgebra theory for transposed Poisson algebras via anti-pre-Lie bialgebras and anti-pre-Lie-Poisson bialgebras
• The $L_{\infty }$-deformations of associative Rota-Baxter algebras and homotopy Rota-Baxter operators
• Totally compatible associative and Lie dialgebras, tridendriform algebras and PostLie algebras
• Rota-Baxter type operators, rewriting systems and Gröbner-Shirshov bases
• Anti-flexible bialgebras
• Curved O-operator systems
• Infinite-dimensional Lie bialgebras via affinization of perm bialgebras and pre-Lie bialgebras
• New splittings of operations of Poisson algebras and transposed Poisson algebras and related algebraic structures
• Anti-dendriform algebras, new splitting of operations and Novikov type algebras
• $O$-operators on associative algebras and associative Yang-Baxter equations
• Quasi-triangular, factorizable Leibniz bialgebras and relative Rota-Baxter operators
• O-operators on associative algebras and dendriform algebras
• $\mathcal{O}$-operators and related structures on Leibniz algebras
• Pre-anti-flexible bialgebra
• Homotopy Rota-Baxter operators, homotopy $\mathcal{O}$-operators and homotopy post-Lie algebras
• Operator forms of nonhomogeneous associative classical Yang-Baxter equation
• Manin triples and quasitriangular structures of Hom-Poisson bialgebras
• A new approach to the bialgebra theory for relative Poisson algebras
• Noncommutative Poisson bialgebras
• On antisymmetric infinitesimal conformal bialgebras
• The category and operad of matching dialgebras