Search engine for discovering works of Art, research articles, and books related to Art and Culture
Javascript must be enabled to continue!
Quasi-pre-Lie bialgebras and twisting of pre-Lie algebras
View through CrossRef
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.
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.
Related Results
Differential Antisymmetric Infinitesimal Bialgebras, Coherent Derivations and Poisson Bialgebras
Differential Antisymmetric Infinitesimal Bialgebras, Coherent Derivations and Poisson Bialgebras
We establish a bialgebra theory for differential algebras, called differential antisymmetric infinitesimal (ASI) bialgebras by generalizing the study of ASI bialgebras to the conte...
Differential graded vertex Lie algebras
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
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
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 logica...
Averaging pre-Lie bialgebras and the related admissible classical Yang-Baxter equations
Averaging pre-Lie bialgebras and the related admissible classical Yang-Baxter equations
In this paper, we initiate the representation theory for averaging pre-Lie algebras, and establish the intrinsic equivalence among matched pairs, Manin triples, and bialgebra struc...
Quantum B-algebras
Quantum B-algebras
Abstract
The concept of quantale was created in 1984 to develop a framework for non-commutative spaces and quantum mechanics with a view toward non-commutative logic...
Relations between L-algebras and other logical algebras
Relations between L-algebras and other logical algebras
In this paper, by considering the notion of L-algebra, we show that there are relations between L-algebras and some of other logical algebras such as residuated lattices, MTL-alge...
Finitely Presented Heyting Algebras
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...