In this paper, we study the relation of the algebraic properties of the higher-order Courant bracket and Dorfman bracket on the direct sum bundle TM⊕∧<sup>p</sup>T*M for an m-dimensional smooth mani...In this paper, we study the relation of the algebraic properties of the higher-order Courant bracket and Dorfman bracket on the direct sum bundle TM⊕∧<sup>p</sup>T*M for an m-dimensional smooth manifold M, and a Lie 2-algebra which is a “categorified” version of a Lie algebra. We prove that the higher-order Courant algebroids give rise to a semistrict Lie 2-algebra, and we prove that the higher-order Dorfman algebroids give rise to a hemistrict Lie 2-algebra. Consequently, there is an isomorphism from the higher-order Courant algebroids to the higher-order Dorfman algebroids as Lie 2-algebras homomorphism.展开更多
In this paper,we give the notion of derivations of Lie 2-algebras using explicit formulas,and construct the associated derivation Lie 3-algebra.We prove that isomorphism classes of non-abelian extensions of Lie 2-alge...In this paper,we give the notion of derivations of Lie 2-algebras using explicit formulas,and construct the associated derivation Lie 3-algebra.We prove that isomorphism classes of non-abelian extensions of Lie 2-algebras are classified by equivalence classes of morphisms from a Lie 2-algebra to a derivation Lie 3-algebra.展开更多
In this article, we introduce the notions of restricted Lie 2-algebras and crossed modules of restricted Lie algebras, and give a series of examples of restricted Lie 2-algebras. We also construct restricted Lie 2-alg...In this article, we introduce the notions of restricted Lie 2-algebras and crossed modules of restricted Lie algebras, and give a series of examples of restricted Lie 2-algebras. We also construct restricted Lie 2-algebras from A（m）-algebras, restricted Leibniz algebras, restricted right-symmetric algebras. Finally, we prove that there is a one-to-one correspondence between strict restricted Lie 2-algebras and crossed modules of restricted Lie algebras.展开更多
A Lie 2-bialgebra is a Lie 2-algebra equipped with a compatible Lie 2-coalgebra structure. In this paper, we give another equivalent description for Lie2-bialgebras by using the structure maps and compatibility condit...A Lie 2-bialgebra is a Lie 2-algebra equipped with a compatible Lie 2-coalgebra structure. In this paper, we give another equivalent description for Lie2-bialgebras by using the structure maps and compatibility conditions. We can use this method to check whether a 2-term direct sum of vector spaces is a Lie 2-bialgebra easily.展开更多
A modified direct method is developed to find finite symmetry groups of nonlinear mathematical physicssystems.Applying the modified direct method to the well-known (2+1)-dimensional BKP equation we get its symmetry.Fu...A modified direct method is developed to find finite symmetry groups of nonlinear mathematical physicssystems.Applying the modified direct method to the well-known (2+1)-dimensional BKP equation we get its symmetry.Furthermore,the exact solutions of (2+1)-dimensional BKP equation are obtained through symmetry analysis.展开更多
We consider the quantum mechanical SU（2） transformation e^2λJzJ±e^-2λJz = e^±2λJ±as if the meaning of squeezing with e^±2λ being squeezing parameter. By studying SU（2） operators （J±,...We consider the quantum mechanical SU（2） transformation e^2λJzJ±e^-2λJz = e^±2λJ±as if the meaning of squeezing with e^±2λ being squeezing parameter. By studying SU（2） operators （J±, Jz） from the point of view of squeezing we find that （J±,Jz） can also be realized in terms of 3-mode bosonic operators. Employing this realization, we find the natural representation （the eigenvectors of J＋ or J-） of the 3-mode squeezing operator e^2λJz. The idea of considering quantum SU（2） transformation as if squeezing is liable for us to obtain the new bosonic operator realization of SU（2） and new squeezing operators.展开更多
This article mainly discusses the direct sum decomposition of type G_2 Lie algebra, which, under such decomposition, is decomposed into a type A_1 simple Lie algebra and one of its modules. Four theorems are given to ...This article mainly discusses the direct sum decomposition of type G_2 Lie algebra, which, under such decomposition, is decomposed into a type A_1 simple Lie algebra and one of its modules. Four theorems are given to describe this module,which could be the direct sum of two or three irreducible modules, or the direct sum of weight modules and trivial modules, or the highest weight module.展开更多
文摘In this paper, we study the relation of the algebraic properties of the higher-order Courant bracket and Dorfman bracket on the direct sum bundle TM⊕∧<sup>p</sup>T*M for an m-dimensional smooth manifold M, and a Lie 2-algebra which is a “categorified” version of a Lie algebra. We prove that the higher-order Courant algebroids give rise to a semistrict Lie 2-algebra, and we prove that the higher-order Dorfman algebroids give rise to a hemistrict Lie 2-algebra. Consequently, there is an isomorphism from the higher-order Courant algebroids to the higher-order Dorfman algebroids as Lie 2-algebras homomorphism.
基金supported by National Natural Science Foundation of China(Grant Nos.11026046,11101179,10971071)Doctoral Fund of Ministry of Education of China(Grant No.20100061120096)the Fundamental Research Funds for the Central Universities(Grant No.200903294)
文摘In this paper,we give the notion of derivations of Lie 2-algebras using explicit formulas,and construct the associated derivation Lie 3-algebra.We prove that isomorphism classes of non-abelian extensions of Lie 2-algebras are classified by equivalence classes of morphisms from a Lie 2-algebra to a derivation Lie 3-algebra.
基金Supported by ZJNSF(Grant Nos.LY17A010015 and LZ14A010001)NNSF(Grant No.11171296)
文摘In this article, we introduce the notions of restricted Lie 2-algebras and crossed modules of restricted Lie algebras, and give a series of examples of restricted Lie 2-algebras. We also construct restricted Lie 2-algebras from A（m）-algebras, restricted Leibniz algebras, restricted right-symmetric algebras. Finally, we prove that there is a one-to-one correspondence between strict restricted Lie 2-algebras and crossed modules of restricted Lie algebras.
文摘A Lie 2-bialgebra is a Lie 2-algebra equipped with a compatible Lie 2-coalgebra structure. In this paper, we give another equivalent description for Lie2-bialgebras by using the structure maps and compatibility conditions. We can use this method to check whether a 2-term direct sum of vector spaces is a Lie 2-bialgebra easily.
基金National Natural Science Foundation of China under Grant Nos.90203001,90503006,0475055,and 10647112the Foundation of Donghua University
文摘A modified direct method is developed to find finite symmetry groups of nonlinear mathematical physicssystems.Applying the modified direct method to the well-known (2+1)-dimensional BKP equation we get its symmetry.Furthermore,the exact solutions of (2+1)-dimensional BKP equation are obtained through symmetry analysis.
基金supported by the National Natural Science Foundation of China(Grant Nos.11175113 and 11275123)the Key Project of Natural Science Fund of Anhui Province,China(Grant No.KJ2013A261)
文摘We consider the quantum mechanical SU（2） transformation e^2λJzJ±e^-2λJz = e^±2λJ±as if the meaning of squeezing with e^±2λ being squeezing parameter. By studying SU（2） operators （J±, Jz） from the point of view of squeezing we find that （J±,Jz） can also be realized in terms of 3-mode bosonic operators. Employing this realization, we find the natural representation （the eigenvectors of J＋ or J-） of the 3-mode squeezing operator e^2λJz. The idea of considering quantum SU（2） transformation as if squeezing is liable for us to obtain the new bosonic operator realization of SU（2） and new squeezing operators.
基金The CLRPF(17pzxmyb10) of Guangdong Peizheng College
文摘This article mainly discusses the direct sum decomposition of type G_2 Lie algebra, which, under such decomposition, is decomposed into a type A_1 simple Lie algebra and one of its modules. Four theorems are given to describe this module,which could be the direct sum of two or three irreducible modules, or the direct sum of weight modules and trivial modules, or the highest weight module.
