Coquasitriangular infinitesimal BiHom-bialgebras and related structures

Abstract

The aim of this paper is to investigate representation theory of infinitesimal (BiHom-)bialgebras of any weight λ (abbr. λ-inf(BH)-bialgebras). Firstly, inspired by the well-known Majid-Radford's bosonization theory in Hopf algebra theory, we present a class of λ-inf(BH)bialgebras, named λ-inf(BH)-biproduct bialgebras, consisting of an inf(BH)-product algebra structure and an inf(BH)-coproduct coalgebra structure, which induces a structure of a λ-inf(BH)-Hopf bimodule over a λ-inf(BH)-bialgebra. Secondly, we explore relationships among λ-inf(BH)-Hopf bimodules, λ-Rota-Baxter (BiHom-)bimodules, (BiHom-)dendriform bimodules and (BiHom-)pre-Lie bimodules. Finally, we provide two kinds of general Gelfand-Dorfman theorems related to BiHom-Novikov algebras. Contents 26 Acknowledgment 29 References 29

Transposed BiHom-Poisson algebras

“…Roughly speaking, a BiHom-associative algebra (or Lie algebra) is an algebra (or Lie algebra) such that the associativity (or Jacobi condition) is twisted by two (commuting) endomorphisms, for details see [10], which can be seen as an extension of Hom-type algebra [13] arising in quasi-deformations of Lie algebras of vector fields. Now there are so many research related to BiHom-type algebras, see refs [5,11,12,[15][16][17][18][19][20][21][23][24][25][26][27][28]. In [21], the authors introduced the notion of BiHom-Poisson algebras and gave a necessary and sufficient condition under which BiHom-Novikov-Poisson algebras (which are twisted generalizations of Novikov-Poisson algebras [30] and Hom-Novikov-Poisson algebras [31]) give rise to BiHom-Poisson algebras.…”