In this paper,we introduce non-abelian cohomology groups and classify the nonabelian extensions of Rota-Baxter pre-Lie algebras in terms of non-abelian cohomology groups.Next,we explore the inducibility of pairs of au...In this paper,we introduce non-abelian cohomology groups and classify the nonabelian extensions of Rota-Baxter pre-Lie algebras in terms of non-abelian cohomology groups.Next,we explore the inducibility of pairs of automorphisms and derive the analog Wells exact sequences under the circumstance of Rota-Baxter pre-Lie algebras.Finally,we discuss the inducibility problem of pairs of automorphisms about an abelian extensions of Rota-Baxter pre-Lie algebras.展开更多
In this paper, we find Hall-Shirshov type bases for free pre-Lie algebras. We show that Segal＇s basis of a free pre-Lie algebra is a type of these bases. We give a nonassociative GrSbner-Shirshov basis S for a free p...In this paper, we find Hall-Shirshov type bases for free pre-Lie algebras. We show that Segal＇s basis of a free pre-Lie algebra is a type of these bases. We give a nonassociative GrSbner-Shirshov basis S for a free pre-Lie algebra such that Irr（S） is a monomial basis （called normal words） of a free pre-Lie algebra, where Irr（S） is the set of all nonassociative words, not containing maximal nonassociative words of polynomials from S. We establish the Composition-Diamond lemma for free pre-Lie algebras over the basis of normal words and the degree breadth lexicographic ordering.展开更多
With the help of Rota-Baxter operators and Grobner-Shirshov bases,we prove that every pre-Lie algebra can be injectively embedded into its universal enveloping preassociative algebra.
A compatible Lie algebra is a pair of Lie algebras such that any linear combination of the two Lie brackets is a Lie bracket. We construct a bialgebra theory of compatible Lie Mgebras as an analogue of a piiLie bialge...A compatible Lie algebra is a pair of Lie algebras such that any linear combination of the two Lie brackets is a Lie bracket. We construct a bialgebra theory of compatible Lie Mgebras as an analogue of a piiLie bialgebra. They can also be regarded as a ＂compatible version＂ of Lie bialgebras, that is, a pair of Lie biaJgebras such that any linear combination of the two Lie bialgebras is still a Lie bialgebra. Many properties of compatible Lie bialgebras as the ＂compatible version＂ of the corresponding properties of Lie biaJgebras are presented. In particular, there is a coboundary compatible Lie bialgebra theory with a construction from the classical Yang-Baxter equation in compatible Lie algebras as a combination of two classical Yang-Baxter equations in lAe algebras. FUrthermore, a notion of compatible pre-Lie Mgebra is introduced with an interpretation of its close relation with the classical Yang-Baxter equation in compatible Lie a/gebras which leads to a construction of the solutions of the latter. As a byproduct, the compatible Lie bialgebras fit into the framework to construct non-constant solutions of the classical Yang-Baxter equation given by Golubchik and Sokolov.展开更多
基金Supported by the National Natural Science Foundation of China(Grant No.12161013)the School-Level Student Research Project of Guizhou University of Finance and Economics(Grant No.2024ZXSY239).
文摘In this paper,we introduce non-abelian cohomology groups and classify the nonabelian extensions of Rota-Baxter pre-Lie algebras in terms of non-abelian cohomology groups.Next,we explore the inducibility of pairs of automorphisms and derive the analog Wells exact sequences under the circumstance of Rota-Baxter pre-Lie algebras.Finally,we discuss the inducibility problem of pairs of automorphisms about an abelian extensions of Rota-Baxter pre-Lie algebras.
文摘In this paper, we find Hall-Shirshov type bases for free pre-Lie algebras. We show that Segal＇s basis of a free pre-Lie algebra is a type of these bases. We give a nonassociative GrSbner-Shirshov basis S for a free pre-Lie algebra such that Irr（S） is a monomial basis （called normal words） of a free pre-Lie algebra, where Irr（S） is the set of all nonassociative words, not containing maximal nonassociative words of polynomials from S. We establish the Composition-Diamond lemma for free pre-Lie algebras over the basis of normal words and the degree breadth lexicographic ordering.
基金supported by the Austrian Science Foundation FWF grant P28079.
文摘With the help of Rota-Baxter operators and Grobner-Shirshov bases,we prove that every pre-Lie algebra can be injectively embedded into its universal enveloping preassociative algebra.
基金Supported by National Natural Science Foundation of China under Grant Nos.11271202,11221091,11425104Specialized Research Fund for the Doctoral Program of Higher Education under Grant No.20120031110022
文摘A compatible Lie algebra is a pair of Lie algebras such that any linear combination of the two Lie brackets is a Lie bracket. We construct a bialgebra theory of compatible Lie Mgebras as an analogue of a piiLie bialgebra. They can also be regarded as a ＂compatible version＂ of Lie bialgebras, that is, a pair of Lie biaJgebras such that any linear combination of the two Lie bialgebras is still a Lie bialgebra. Many properties of compatible Lie bialgebras as the ＂compatible version＂ of the corresponding properties of Lie biaJgebras are presented. In particular, there is a coboundary compatible Lie bialgebra theory with a construction from the classical Yang-Baxter equation in compatible Lie algebras as a combination of two classical Yang-Baxter equations in lAe algebras. FUrthermore, a notion of compatible pre-Lie Mgebra is introduced with an interpretation of its close relation with the classical Yang-Baxter equation in compatible Lie a/gebras which leads to a construction of the solutions of the latter. As a byproduct, the compatible Lie bialgebras fit into the framework to construct non-constant solutions of the classical Yang-Baxter equation given by Golubchik and Sokolov.
