Drinfel’d double for monoidal Hom-Hopf algebras

Abstract

We mainly construct a bicrossproduct for a finite-dimensional monoidal Hom-Hopf algebra (H, α), generalizing Majid's bicrossproduct. Naturally, the Hom-type bicrossproduct leads to the Drinfel'd double (H op H * , α⊗(α −1) *) with a quasitriangular structure R satisfying the quantum Hom-Yang-Baxter equations.

The Drinfel'd codouble constuction for monoidal Hom-Hopf algebra

“…Finally, we will recall the bicrossproduct structure of two monoidal Hom-Hopf algebras, which is an important result in [16]. In this paper we call it left bicrossproduct.…”

“…Hom-structures (Lie algebras, algebras, coalgebras, and Hopf algebras) have been intensively investigated in the literature recently (see [3,5,6]), In [2], Caenepeel and Goyvaerts illustrated from the point of view of monoidal categories and introduced monoidal Hom-Hopf algebras. Soon afterwards, many of classical results in Hopf algebra theory can be generalized to the monoidal Hom-Hopf algebras (see [8,16]).…”

“…Let (A, α) and (H, β) be two monoidal Hom-Hopf algebras. In section 2, we will introduce the definition of an another bicrossproduct (A ⊲⊳ H, α ⊗ β) (called right bicrossproduct in this paper) which is slightly different from one in [16], and the sufficient and necessary conditions for the right bicrossproduct to be a monoidal Hom-Hopf algebra are given, generalizing the structure of the Majid's bicrossproduct defined in [10]. In section 3, we will give a class of right bicrossproduct monoidal Hom-Hopf algebras.…”

A new approach to Rota–Baxter coalgebras

“…Rota-Baxter algebras were introduced in [18] in the context of differential operators on commutative Banach algebras. At present, Rota-Baxter algebras are a useful tool applied in many branches of mathematics, such as combinatorics, Loday type algebras, pre-Lie algebras and pre-Poisson algebras [1,2,11], multiple zeta values [7,8], quantum field theory [4] and other mathematical objects [9,12,13,15,19,28,29]. Rota-Baxter coalgebras [14] are the dual structures to Rota-Baxter algebras.…”