3-Lie bialgebras

3-Lie algebras have close relationships with many important fields in mathematics and mathematical physics. This article concerns 3-Lie algebras. The concepts of 3-Lie coalgebras and 3-Lie bialgebras are given. The structures of such categories of algebras and the relationships with 3-Lie algebras are studied. And the classification of 4-dimensional 3-Lie coalgebras and 3-dimensional 3-Lie bialgebras over an algebraically closed field of characteristic zero are provided.     

DUAL ASPECTS OF THE QUASITRIANGULAR BIALGEBRAS AND THE BRAIDED BIALGEBRAS

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对合导子构造的3-李双代数与3-Pre-李代数

研究具有对合导子的3-李代数的结构,证明了具有对合导子的m-维3-李代数A存在相容的3-Pre-李代数,且在2m-维半直积3-李代数A■_(ad~*)A~*上存在局部上循环3-李双代数结构.利用对合导子构造了3-李代数A■_(ad~*)A~*上的3-李Yang-Baxter方程的解和一类3-Pre-李代数,并构造了8-维和10-维局部上循环3-李双代数.     

关于MH中的Lie双代数和余Poisson-Hopf代数

本文研究余三角Ｈｏｐｆ代数余模范畴中的Ｌｉｅ双代数和余ＰｏｉｓｓｏｎＨｏｐｆ代数,我们主要讨论余三角Ｈｏｐｆ代数余模范畴中的Ｌｉｅ双代数和余Ｐｏｉｓｓｏｎ－Ｈｏｐｆ代数之间的关系。     

3-李代数的辛结构

研究了3-李代数和度量3-李代数的辛结构.对任意3-李代数L,构造了无限多个度量辛3-李代数.证明了度量3-李代数(A,B)是度量辛3-李代数的充要条件,即存在可逆导子D,使得D∈Der_B(A).同时证明了每一个度量辛3-李代数(A,B,ω)是度量辛3-李代数(A,B,ω)的T_θ~*-扩张.最后,利用度量辛3-李代数经过特殊导子的双扩张得到了新的度量辛3-李代数.     

3-Lie bialgebras and 3-Lie classical Yang–Baxter equations in low dimensions

In this paper, we give low-dimensional examples of local cocycle 3-Lie bialgebras and double construction 3-Lie bialgebras which were introduced in the study of the classical Yang–Baxter equation and Manin triples for 3-Lie algebras. We give an explicit and practical formula to compute the skew-symmetric solutions of the 3-Lie classical Yang–Baxter equation (CYBE). As an illustration, we obtain all skew-symmetric solutions of the 3-Lie CYBE in complex 3-Lie algebras of dimensions 3 and 4 and then the induced local cocycle 3-Lie bialgebras. On the other hand, we classify the double construction 3-Lie bialgebras for complex 3-Lie algebras in dimensions 3 and 4 and then give the corresponding eight-dimensional pseudo-metric 3-Lie algebras.     

T*-extension of a 3-Lie algebra

We introduce a new technique called T*-extension, of constructing a metric 3-Lie algebra out of an arbitrary 3-Lie algebra, and explore all possible metrics and corresponding signatures on this resulting metric 3-Lie algebra.     

Dual Coalgebras of Jordan Bialgebras and Superalgebras

W. Michaelis showed for Lie bialgebras that the dual coalgebra of a Lie algebra is a Lie bialgebra. In the present article we study an analogous question in the case of Jordan bialgebras. We prove that a simple infinite-dimensional Jordan superalgebra of vector type possesses a nonzero dual coalgebra. Thereby, we demonstrate that the hypothesis formulated by W. Michaelis for Lie coalgebras fails in the case of Jordan supercoalgebras.     

Structures of Malcev Bialgebras on a Simple Non-Lie Malcev Algebra

In this work we consider Malcev bialgebras. We describe all structures of a Malcev bialgebra on a simple non-Lie Malcev algebra.     

3-Lie代数的次理想

主要研究3-Lie代数的子代数及次理想的结构.证明了由次理想生成的子代数不一定是次理想.给出了由次理想生成的子代数是次理想的充要条件.最后研究了次理想和子代数的G_n-对之间的关系.     

3-Lie Algebras Realized by Cubic Matrices

Associative multiplications of cubic matrices are provided. The N3-dimensional 3-Lie algebras are realized by cubic matrices, and structures of the 3-Lie algebras are studied.     

The classical Yang-Baxter equation on alternative algebras: The alternative D-bialgebra structure on Cayley-Dickson matrix algebras

We consider the Yang-Baxter equations on alternative algebras and prove that the bialgebras induced by the solutions to these equations are alternative D-bialgebras. We describe the alternative D-bialgebra structure on Cayley-Dickson matrix algebras.     

The Realization of 4-dimensional 3-Lie Algebras

In this paper, we mainly investigate the realization of 3-Lie algebras from a family of Lie algebras. We prove the realization theorem, offer a concrete example realizing all type of 4-dimensional 3-Lie algebras, and also give some properties about semi-simple n-Lie algebras.     

4维3-Lie代数的实现

In this paper, we mainly investigate the realization of 3-Lie algebras from a family of Lie algebras. We prove the realization theorem, offer a concrete example realizing all type of 4-dimensional 3-Lie algebras, and also give some properties about semi-simple n-Lie algebras.     

(H,R)-Lie coalgebras and (H,R)-Lie bialgebras

Given an (H,R)-Lie coalgebra Γ, we construct (H,R _T )-Lie coalgebra Γ^T through a right cocycle T, where (H,R) is a triangular Hopf algebra, and prove that there exists a bijection between the set of (H,R)-Lie coalgebras and the set of ordinary Lie coalgebras. We also show that if (L, [, ], Δ, R) is an (H,R)-Lie bialgebra of an ordinary Lie algebra then (L ^T , [, ], Δ_T, R _T ) is an (H,R _T )-Lie bialgebra of an ordinary Lie algebra.     

具有β(L)=m-3的m-维非交换3-李代数的分类*

对特征零域F上具有β(L)=m-3的m-维非交换3-李代数的结构进行了分类, 且给出了每一类3-李代数的具体乘法结构.证明了满足β(L)=m-3且中心含在导代数的非交换3-李代数的维数小于等于11,大于等于5, 且导代数维数等于1时的3-李代数仅有两类, 导代数维数等于2 时有24类.     

3-李代数的广义导子

给出了3-李代数的广义导子、拟导子、拟型心的定义,研究了他们之间的结构关系,并对具有极大对角环面的3-李代数的拟导子和拟型心结构进行了系统的研究.证明了(1)广义导代数GDer(A)可以分解成拟导子代数QDer(A)和拟型心QΓ(A)的直和;(2)3-李代数A的拟导子可以扩张成一个具有较大维数的3-李代数的导子;(3)拟导子代数QDer(A)包含在拟型心的正规化子中,表示为[QDer(A),QΓ(A)]?QΓ(A);(4)如果A包含极大对角环面T,那么QDer(A)和Qr(A)是T的对角模,也就是(T,T)半单地作用在QDer(A)和QΓ(A)上.     

Dual Lie Bialgebra Structures of W -algebra W (2, 2) Type

In the present paper, we investigate the dual Lie coalgebras of the centerless W(2, 2) algebra by studying the maximal good subspaces. Based on this, we construct the dual Lie bialgebra structures of the centerless W(2, 2) Lie bialgebra. As by-products, four new infinite dimensional Lie algebras are obtained.