Jordan D-bialgebras were introduced by Zhelyabin.In this paper,we use a new approach to study Jordan D-bialgebras by a new notion of the dual representation of the regular representation of a Jordan algebra.Motivated ...Jordan D-bialgebras were introduced by Zhelyabin.In this paper,we use a new approach to study Jordan D-bialgebras by a new notion of the dual representation of the regular representation of a Jordan algebra.Motivated by the essential connection between Lie bialgebras and Manin triples,we give an explicit proof of the equivalence between Jordan D-bialgebras and a class of special Jordan-Manin triples called double constructions of pseudo-euclidean Jordan algebras.We also show that a Jordan D-bialgebra leads to the Jordan Yang-Baxter equation under the coboundary condition and an antisymmetric nondegenerate solution of the Jordan Yang-Baxter equation corresponds to an antisymmetric bilinear form,which we call a Jordan symplectic form on Jordan algebras.Furthermore,there exists a new algebra structure called pre-Jordan algebra on Jordan algebras with a Jordan symplectic form.展开更多
In this paper, we introduce a notion of J-dendriform algebra with two operations as a Jordan algebraic analogue of a dendriform algebra such that the antieommutator of the sum of the two operations is a Jordan algebra...In this paper, we introduce a notion of J-dendriform algebra with two operations as a Jordan algebraic analogue of a dendriform algebra such that the antieommutator of the sum of the two operations is a Jordan algebra. A dendriform algebra is a J-dendriform algebra. Moreover, J-dendriform algebras fit into a commutative diagram which extends the relationships among associative, Lie, and Jordan algebras. Their relations with some structures such as Rota-Baxter operators, classical Yang-Baxter equation, and bilinear forms are given.展开更多
Abstract In this paper, the super t^-operators of Jordan superalgebras are introduced and the solutions of super Jordan Yang-Baxter equation are discussed by super б-operators. Then pre-Jordan superalgebras are studi...Abstract In this paper, the super t^-operators of Jordan superalgebras are introduced and the solutions of super Jordan Yang-Baxter equation are discussed by super б-operators. Then pre-Jordan superalgebras are studied as the algebraic structure behind the super б-operators. Moreover, the relations among Jordan superalgebras, pre-Jordan superalgebras, and dendriform superalgebras are established. Keywords Super б-operator, dendriform superalgebra, pre-Jordan superalgebra展开更多
文摘Jordan D-bialgebras were introduced by Zhelyabin.In this paper,we use a new approach to study Jordan D-bialgebras by a new notion of the dual representation of the regular representation of a Jordan algebra.Motivated by the essential connection between Lie bialgebras and Manin triples,we give an explicit proof of the equivalence between Jordan D-bialgebras and a class of special Jordan-Manin triples called double constructions of pseudo-euclidean Jordan algebras.We also show that a Jordan D-bialgebra leads to the Jordan Yang-Baxter equation under the coboundary condition and an antisymmetric nondegenerate solution of the Jordan Yang-Baxter equation corresponds to an antisymmetric bilinear form,which we call a Jordan symplectic form on Jordan algebras.Furthermore,there exists a new algebra structure called pre-Jordan algebra on Jordan algebras with a Jordan symplectic form.
基金Acknowledgements This work was supported in part by the National Natural Science Foundation of China （Grant Nos. 10621101, 10921061）, the National Key Basic Research Development Project （2006CB805905）, and the Specialized Research Fund for the Doctoral Program （200800550015）.
文摘In this paper, we introduce a notion of J-dendriform algebra with two operations as a Jordan algebraic analogue of a dendriform algebra such that the antieommutator of the sum of the two operations is a Jordan algebra. A dendriform algebra is a J-dendriform algebra. Moreover, J-dendriform algebras fit into a commutative diagram which extends the relationships among associative, Lie, and Jordan algebras. Their relations with some structures such as Rota-Baxter operators, classical Yang-Baxter equation, and bilinear forms are given.
文摘Abstract In this paper, the super t^-operators of Jordan superalgebras are introduced and the solutions of super Jordan Yang-Baxter equation are discussed by super б-operators. Then pre-Jordan superalgebras are studied as the algebraic structure behind the super б-operators. Moreover, the relations among Jordan superalgebras, pre-Jordan superalgebras, and dendriform superalgebras are established. Keywords Super б-operator, dendriform superalgebra, pre-Jordan superalgebra
