In this paper, by using the Ringel-Hall algebra method, we prove that the set of the skew-commutator relations of quantum root vectors forms a minimal GrSbner- Shirshov basis for the quantum groups of Dynkin type. As ...In this paper, by using the Ringel-Hall algebra method, we prove that the set of the skew-commutator relations of quantum root vectors forms a minimal GrSbner- Shirshov basis for the quantum groups of Dynkin type. As an application, we give an explicit basis for the types E7 and Dn.展开更多
Motivated by the concept of matching Rota-Baxter algebras arising from polarized associative Yang-Baxter equations and Volterra integral equations,we introduce the notion of a matching Rota-Baxter system,which general...Motivated by the concept of matching Rota-Baxter algebras arising from polarized associative Yang-Baxter equations and Volterra integral equations,we introduce the notion of a matching Rota-Baxter system,which generalizes the Rota-Baxter system proposed by Brzezinski.We show that this notion is also related to Yang-Baxter pairs and to matching pre-Lie algebras.We then modify the definition of matching Rota-Baxter systems by adding a curvature term,and make a connection with matching pre-Lie algebras and with compatible associative algebras.Furthermore,we study matching Rota-Baxter systems on a dendriform algebra and show how they induce matching quadri-algebra structures.Finally,we give a linear basis of free matching Rota-Baxter system by Grobner-Shirshov bases methods.展开更多
In this article,we construct free centroid hom-associative algebras and free centroid hom-Lie algebras.We also construct some other relatively free centroid hom-associative algebras by applying the Gr?bner-Shirshov ba...In this article,we construct free centroid hom-associative algebras and free centroid hom-Lie algebras.We also construct some other relatively free centroid hom-associative algebras by applying the Gr?bner-Shirshov basis theory for(unital)centroid hom-associative algebras.Finally,we prove that the"Poincaré-Birkhoff-Witt theorem"holds for certain type of centroid hom-Lie algebras over a field of characteristic 0,namely,every centroid hom-Lie algebra such that the eigenvectors of the mapβlinearly generates the whole algebra can be embedded into its universal enveloping centroid hom-associative algebra,and the linear basis of the universal enveloping algebra does not depend on the multiplication table of the centroid hom-Lie algebra under consideration.展开更多
In this paper, by using Gr bner-Shirshov bases theories, we prove that each countably generated associative differential algebra （resp., associative λ-algebra, associa- tive Ω-differential algebra） can be embedded...In this paper, by using Gr bner-Shirshov bases theories, we prove that each countably generated associative differential algebra （resp., associative λ-algebra, associa- tive Ω-differential algebra） can be embedded into a simple 2-generated associative differ- ential algebra （resp., associative Ωalgebra, associative λ-differential algebra）.展开更多
Differential difference algebras are generalizations of polynomial algebras, quantum planes, and Ore extensions of automorphism type and of derivation type. In this paper, we investigate the Gelfand-Kirillov dimension...Differential difference algebras are generalizations of polynomial algebras, quantum planes, and Ore extensions of automorphism type and of derivation type. In this paper, we investigate the Gelfand-Kirillov dimension of a finitely generated module over a differential difference algebra through a computational method： Grobner-Shirshov basis method. We develop the GrSbner-Shirshov basis theory of differential difference al- gebras, and of finitely generated modules over differential difference algebras, respectively. Then, via GrSbner-Shirshov bases, we give algorithms for computing the Gelfand-Kirillov dimensions of cyclic modules and finitely generated modules over differential difference algebras.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.
基金Supported by the National Natural Science Foundation of China （11061033）.Acknowledgements. Part of this work is done during the corresponding author＇s visiting the Stuttgart University with the support of China Scholarship Council. With this opportunity, he expresses his gratefulness to Professor Steffen Koenig and the Institute of Algebra and Number Theory of Stuttgart University and the China Scholarship Council.
文摘In this paper, by using the Ringel-Hall algebra method, we prove that the set of the skew-commutator relations of quantum root vectors forms a minimal GrSbner- Shirshov basis for the quantum groups of Dynkin type. As an application, we give an explicit basis for the types E7 and Dn.
基金supported by the National Natural Science Foundation of China(12161013,12101316)supported by the Belt and Road Innovative Talents Exchange Foreign Experts Project(DL2023014002L).
文摘Motivated by the concept of matching Rota-Baxter algebras arising from polarized associative Yang-Baxter equations and Volterra integral equations,we introduce the notion of a matching Rota-Baxter system,which generalizes the Rota-Baxter system proposed by Brzezinski.We show that this notion is also related to Yang-Baxter pairs and to matching pre-Lie algebras.We then modify the definition of matching Rota-Baxter systems by adding a curvature term,and make a connection with matching pre-Lie algebras and with compatible associative algebras.Furthermore,we study matching Rota-Baxter systems on a dendriform algebra and show how they induce matching quadri-algebra structures.Finally,we give a linear basis of free matching Rota-Baxter system by Grobner-Shirshov bases methods.
基金the grant of Guangzhou Civil Aviation College(Grant No.22X0430)the RAS Fundamental Research Program(Grant No.FWNF-2022-0002)+2 种基金the NNSF of China(Grant Nos.11571121,12071156)the NNSF of China(Grant No.12101248)the China Postdoctoral Science Foundation(Grant No.2021M691099)。
文摘In this article,we construct free centroid hom-associative algebras and free centroid hom-Lie algebras.We also construct some other relatively free centroid hom-associative algebras by applying the Gr?bner-Shirshov basis theory for(unital)centroid hom-associative algebras.Finally,we prove that the"Poincaré-Birkhoff-Witt theorem"holds for certain type of centroid hom-Lie algebras over a field of characteristic 0,namely,every centroid hom-Lie algebra such that the eigenvectors of the mapβlinearly generates the whole algebra can be embedded into its universal enveloping centroid hom-associative algebra,and the linear basis of the universal enveloping algebra does not depend on the multiplication table of the centroid hom-Lie algebra under consideration.
文摘In this paper, by using Gr bner-Shirshov bases theories, we prove that each countably generated associative differential algebra （resp., associative λ-algebra, associa- tive Ω-differential algebra） can be embedded into a simple 2-generated associative differ- ential algebra （resp., associative Ωalgebra, associative λ-differential algebra）.
文摘Differential difference algebras are generalizations of polynomial algebras, quantum planes, and Ore extensions of automorphism type and of derivation type. In this paper, we investigate the Gelfand-Kirillov dimension of a finitely generated module over a differential difference algebra through a computational method： Grobner-Shirshov basis method. We develop the GrSbner-Shirshov basis theory of differential difference al- gebras, and of finitely generated modules over differential difference algebras, respectively. Then, via GrSbner-Shirshov bases, we give algorithms for computing the Gelfand-Kirillov dimensions of cyclic modules and finitely generated modules over differential difference algebras.
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.
