Javascript must be enabled to continue!

Quasi-pre-Lie bialgebras and twisting of pre-Lie algebras

Given a (quasi-)twilled pre-Lie algebra, we first construct a differential graded Lie algebra ([Formula: see text]-algebra). Then we study the twisting theory of (quasi-)twilled pre-Lie algebras and show that the result of the twisting by a linear map on a (quasi-)twilled pre-Lie algebra is also a (quasi-)twilled pre-Lie algebra if and only if the linear map is a solution of the Maurer–Cartan equation of the associated differential graded Lie algebra ([Formula: see text]-algebra). In particular, the relative Rota–Baxter operators (twisted relative Rota–Baxter operators) on pre-Lie algebras are solutions of the Maurer–Cartan equation of the differential graded Lie algebra ([Formula: see text]-algebra) associated to the certain quasi-twilled pre-Lie algebra. Finally, we use the twisting theory of (quasi-)twilled pre-Lie algebras to study quasi-pre-Lie bialgebras. Moreover, we give a construction of quasi-pre-Lie bialgebras through symplectic Lie algebras, which is parallel to that a Cartan [Formula: see text]-form on a semi-simple Lie algebra gives a quasi-Lie bialgebra.

World Scientific Pub Co Pte Ltd

Qi Wang Jiefeng Liu

Journal of Algebra and Its Applications

2023

Title: Quasi-pre-Lie bialgebras and twisting of pre-Lie algebras

Description:

Given a (quasi-)twilled pre-Lie algebra, we first construct a differential graded Lie algebra ([Formula: see text]-algebra).

Then we study the twisting theory of (quasi-)twilled pre-Lie algebras and show that the result of the twisting by a linear map on a (quasi-)twilled pre-Lie algebra is also a (quasi-)twilled pre-Lie algebra if and only if the linear map is a solution of the Maurer–Cartan equation of the associated differential graded Lie algebra ([Formula: see text]-algebra).

In particular, the relative Rota–Baxter operators (twisted relative Rota–Baxter operators) on pre-Lie algebras are solutions of the Maurer–Cartan equation of the differential graded Lie algebra ([Formula: see text]-algebra) associated to the certain quasi-twilled pre-Lie algebra.

Finally, we use the twisting theory of (quasi-)twilled pre-Lie algebras to study quasi-pre-Lie bialgebras.

Moreover, we give a construction of quasi-pre-Lie bialgebras through symplectic Lie algebras, which is parallel to that a Cartan [Formula: see text]-form on a semi-simple Lie algebra gives a quasi-Lie bialgebra.

Back

Related Results

Differential graded vertex Lie algebras

This is the continuation of the study of differential graded (dg) vertex algebras defined in our previous paper [Caradot et al., “Differential graded vertex operator algebras and t...

Coherent categorical structures for Lie bialgebras, Manin triples, classical r-matrices and pre-Lie algebras

Abstract The broadly applied notions of Lie bialgebras, Manin triples, classical r-matrices and ...

Some Results on Quasi MV-Algebras and Perfect Quasi MV-Algebras

Abstract Quasi MV-algebras are a generalization of MV-algebras and they are motivated by the investigation of the structure of quantum logical gates. In the first part, w...

Weak pseudo-BCK algebras

Abstract In this paper we define and study the weak pseudo-BCK algebras as generalizations of weak BCK-algebras, extending some results given by Cı⃖rulis for weak BC...

Finitely Presented Heyting Algebras

In this paper we study the structure of finitely presented Heyting<br />algebras. Using algebraic techniques (as opposed to techniques from proof-theory) we show that every s...

3-Hom–Lie Yang–Baxter Equation and 3-Hom–Lie Bialgebras

In this paper, we first introduce the notion of a 3-Hom–Lie bialgebra and give an equivalent description of the 3-Hom–Lie bialgebras, the matched pairs and the Manin triples of 3-H...

Realizations of 3-Lie algebras

3-Lie algebras have close relationships with many important fields in mathematics and mathematical physics. In this paper, we provide a construction of 3-Lie algebras in terms of L...

Central invariants and enveloping algebras of braided Hom-Lie algebras

Let (H,?) be a monoidal Hom-Hopf algebra and HH HYD the Hom-Yetter-Drinfeld category over (H,?). Then in this paper, we first introduce the definition of braided Hom-Lie alge...

Email:
Password:

Email: