Double constructions of biHom-Frobenius algebras

Abstract

This paper addresses a Hom-associative algebra built as a direct sum of a given Hom-associative algebra $(\mathcal{A},\cdot ,\alpha )$ and its dual $({\mathcal{A}}^{\ast },\circ ,{\alpha }^{\ast }),$ endowed with a non-degenerate symmetric bilinear form $\mathcal{B},$ where $\cdot$ and $\circ$ are the products defined on $\mathcal{A}$ and ${\mathcal{A}}^{\ast },$ respectively, and $\alpha$ and ${\alpha }^{\ast }$ stand for the corresponding algebra homomorphisms. Such a double construction, also called Hom-Frobenius algebra, is interpreted in terms of an infinitesimal Hom-bialgebra. The same procedure is applied to characterize the double construction of biHom-associative algebras, also called biHom-Frobenius algebra. Finally, a double construction of Hom-dendriform algebras, also called double construction of Connes cocycle or symplectic Hom-associative algebra, is performed. Besides, the concept of biHom-dendriform algebras is introduced and discussed. Their bimodules and matched pairs are also constructed, and related relevant properties are given.