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
A theorem of Maurer-Cartan type for Lie algebroids is presented. Suppose that any vector subbundle of a Lie algebroid is called interior differential system (IDS) for that Lie algebroid. A theorem of Frobenius type is...A theorem of Maurer-Cartan type for Lie algebroids is presented. Suppose that any vector subbundle of a Lie algebroid is called interior differential system (IDS) for that Lie algebroid. A theorem of Frobenius type is obtained. Extending the classical notion of exterior diffential system (EDS) to Lie algebroids, a theorem of Cartan type is obtained.展开更多
In this paper,we study the algebraic properties of the higher analogues of Courant algebroid structures on the direct sum bundle TM⊕∧nT*M for an m-dimensional manifold.As an application,we revisit Nambu-Poisson stru...In this paper,we study the algebraic properties of the higher analogues of Courant algebroid structures on the direct sum bundle TM⊕∧nT*M for an m-dimensional manifold.As an application,we revisit Nambu-Poisson structures and multisymplectic structures.We prove that the graph of an(n+1)-vector fieldπis closed under the higher-order Dorfman bracket iffπis a Nambu-Poisson structure.Consequently,there is an induced Leibniz algebroid structure on∧nT*M.The graph of an(n+1)-formωis closed under the higher-order Dorfman bracket iffωis a premultisymplectic structure of order n,i.e.,dω=0.Furthermore,there is a Lie algebroid structure on the admissible bundle A∧nT*M.In particular,for a 2-plectic structure,it induces the Lie 2-algebra structure given in(Baez,Hoffnung and Rogers,2010).展开更多
In this paper,we study the structure of nonabelian omni-Lie algebroids.Firstly,taking Lie algebroid(E,[·,·]_(E,ρE))as the starting point,a nonabelian omni-Lie algebroid is defined on direct sum bundle DE⊕J...In this paper,we study the structure of nonabelian omni-Lie algebroids.Firstly,taking Lie algebroid(E,[·,·]_(E,ρE))as the starting point,a nonabelian omni-Lie algebroid is defined on direct sum bundle DE⊕JE,where DE and JE are,respectively,the gauge Lie algebroid and the jet bundle of vector bundle E,and study its properties.Furthermore,it is concluded that the nonabelian omni-Lie algebroid is a trivial deformation of the omni-Lie algebroid,and the nonabelian omni-Lie algebroid is a matched pair of Leibniz algebroids.展开更多
For any regular Lie algebroid A, the kernel K and the image F of its anchor map ρA, together with A itself fit into a short exact sequence, called the Atiyah sequence, of Lie algebroids. We prove that the Atiyah and ...For any regular Lie algebroid A, the kernel K and the image F of its anchor map ρA, together with A itself fit into a short exact sequence, called the Atiyah sequence, of Lie algebroids. We prove that the Atiyah and Todd classes of dg manifolds arising from a regular Lie algebroid respect the Atiyah sequence, i.e.,the Atiyah and Todd classes of A restrict to the Atiyah and Todd classes of the bundle K of Lie algebras on the one hand, and project onto the Atiyah and Todd classes of the integrable distribution F■T_M on the other hand.展开更多
Protobialgebroids include several kinds of algebroid structures such as Lie algebroid,Lie bialgebroid, Lie quasi-bialgebroid, etc. In this paper, the Dirac theories are generalized from Lie bialgebroid to protobialgeb...Protobialgebroids include several kinds of algebroid structures such as Lie algebroid,Lie bialgebroid, Lie quasi-bialgebroid, etc. In this paper, the Dirac theories are generalized from Lie bialgebroid to protobialgebroid. We give the integrable conditions for a maximally isotropic subbundle being a Dirac structure for a protobialgebroid by the notion of a characteristic pair. From the integrable conditions, we found out that the Dirac structure has closed relations with the twisting of a protobialgebroid. At last, some special cases of the Dirac structures for protobialgebroids are discussed.展开更多
Considering the prolongation of a Lie algebroid,the authors introduce Finsler algebroids and present important geometric objects on these spaces.Important endomorphisms like conservative and Barthel,Cartan tensor and ...Considering the prolongation of a Lie algebroid,the authors introduce Finsler algebroids and present important geometric objects on these spaces.Important endomorphisms like conservative and Barthel,Cartan tensor and some distinguished connections like Berwald,Cartan,Chern-Rund and Hashiguchi are introduced and studied.展开更多
For a transitive Lie algebroid A on a connected manifold M and its representation on a vector bundle F,we define a morphism of cohomology groups rk:Hk(A,F)→Hk(Lx,Fx),called the localization map,where Lx is the adjoin...For a transitive Lie algebroid A on a connected manifold M and its representation on a vector bundle F,we define a morphism of cohomology groups rk:Hk(A,F)→Hk(Lx,Fx),called the localization map,where Lx is the adjoint algebra at x∈M.The main result in this paper is that if M is simply connected,or H(LX,FX)is trivial,then T is injective.This means that the Lie algebroid 1-cohomology is totally determined by the 1-cohomology of its adjoint Lie algebra in the above two cases.展开更多
After introducing some of the basic definitions and results from the theory of groupoid and Lie algebroid,we investigate the discrete Lagrangian mechanics from the viewpoint of groupoid theory and give the connection ...After introducing some of the basic definitions and results from the theory of groupoid and Lie algebroid,we investigate the discrete Lagrangian mechanics from the viewpoint of groupoid theory and give the connection betweengroupoids variation and the methods of the first and second discrete variational principles.展开更多
In this paper, the following theorem is obtained: Let w=w(z) be a υ valued algebroid function of finite order ρ>0 for |z|<∞ , then there exists a direction L: arg z=θ 0(0≤θ 0<2π) ...In this paper, the following theorem is obtained: Let w=w(z) be a υ valued algebroid function of finite order ρ>0 for |z|<∞ , then there exists a direction L: arg z=θ 0(0≤θ 0<2π) , such that for any δ>0(δ<π2 ), lim r→∞ log 3) (r,Δ(θ 0,δ),a) log r=ρ with 3υ-1 possible exceptions for any value of a . Where 3) (r,Δ(θ 0,δ),a) is the numbers of distinct zeros with multiplicity ≤3 of w(z)-a in (θ 0,δ)∩{||≤r} .展开更多
Using the Nevanlinna theory of the value distribution of meromorphic functions, we investigate the existence problem of admissible algebroid solutions of generalized complex algebraic differential equations and obtain...Using the Nevanlinna theory of the value distribution of meromorphic functions, we investigate the existence problem of admissible algebroid solutions of generalized complex algebraic differential equations and obtain some results.展开更多
In 1909 the brothers E. and F. Cosserat discovered a new nonlinear group theoretical approach to elasticity (EL), with the only experimental need to measure the EL constants. In a modern framework, they used the nonli...In 1909 the brothers E. and F. Cosserat discovered a new nonlinear group theoretical approach to elasticity (EL), with the only experimental need to measure the EL constants. In a modern framework, they used the nonlinear Spencer sequence instead of the nonlinear Janet sequence for the Lie groupoid defining the group of rigid motions of space. Following H. Weyl, our purpose is to compute for the first time the linear and nonlinear Spencer sequences for the Lie groupoid defining the conformal group of space-time in order to provide the mathematical foundations of both electromagnetism (EM) and gravitation (GR), with the only experimental need to measure the EM and GR constants. With a manifold of dimension n ≥ 3, the difficulty is to deal with the n nonlinear transformations that have been called “elations” by E. Cartan in 1922. Using the fact that dimension n = 4 has very specific properties for the computation of the Spencer cohomology, we also prove that there is no conceptual difference between the (nonlinear) Cosserat EL field or induction equations and the (linear) Maxwell EM field or induction equations. As for gravitation, the dimension n = 4 also allows to have a conformal factor defined everywhere but at the central attractive mass because the inversion law of the isotropy subgroupoid made by second order jets transforms attraction into repulsion. The mathematical foundations of both electromagnetism and gravitation are thus only depending on the structure of the conformal pseudogroup of space-time.展开更多
In 1909 the brothers E. and F. Cosserat discovered a new nonlinear group theoretical approach to elasticity (EL), with the only experimental need to measure the EL constants. In a modern language, their idea has been ...In 1909 the brothers E. and F. Cosserat discovered a new nonlinear group theoretical approach to elasticity (EL), with the only experimental need to measure the EL constants. In a modern language, their idea has been to use the nonlinear Spencer sequence instead of the nonlinear Janet sequence for the Lie groupoid defining the group of rigid motions of space. Following H. Weyl, our purpose is to compute for the first time the nonlinear Spencer sequence for the Lie groupoid defining the conformal group of space-time in order to provide the mathematical foundations of electromagnetism (EM), with the only experimental need to measure the EM constant in vacuum. With a manifold of dimension n, the difficulty is to deal with the n nonlinear transformations that have been called “elations” by E. Cartan in 1922. Using the fact that dimension n=4 has very specific properties for the computation of the Spencer cohomology, we prove that there is thus no conceptual difference between the Cosserat EL field or induction equations and the Maxwell EM field or induction equations. As a byproduct, the well known field/matter couplings (piezzoelectricity, photoelasticity, streaming birefringence, …) can be described abstractly, with the only experimental need to measure the corresponding coupling constants. The main consequence of this paper is the need to revisit the mathematical foundations of gauge theory (GT) because we have proved that EM was depending on the conformal group and not on U(1), with a shift by one step to the left in the physical interpretation of the differential sequence involved.展开更多
In this article, we first investigate the operational properties of algebroid functions. Then we prove two uniqueness theorems for algebroid functions.
Using Nevanlinna theory of the value distribution of meromorphic functions, the author investigates the problem of the growth of solutions of two types of algebraic differential equation and obtains some results.
In this paper, we investigate the normality relationship between algebroid multifunctions and their coefficient functions. We prove that the normality of a k-valued entire algebroid multifunctions family is equivalent...In this paper, we investigate the normality relationship between algebroid multifunctions and their coefficient functions. We prove that the normality of a k-valued entire algebroid multifunctions family is equivalent to their coefficient functions in some conditions. Furthermore, we obtain some new normality criteria for algebroid multifunctions families based on these results. We also provide some examples to expound that some restricted conditions of our main results are necessary.展开更多
In this article,the authors define the derived function of an algeboidal function in the unit disc,prove it is an algabriodal function,and study the order of algebroidal function and that of its derived function in un...In this article,the authors define the derived function of an algeboidal function in the unit disc,prove it is an algabriodal function,and study the order of algebroidal function and that of its derived function in unit circular disc.展开更多
By using the definition of Hausdorff distance, we prove some normality criteria for families of meromorphic algebroid functions. Some examples are given to complement the theory in this article.
Using Ahlfors’ theory of covering surface and a type-function,we confirm the existence theorem of a Borel radius and a T-radius for the algebroidal function dealing with multiple values in the unit disc,which briefly...Using Ahlfors’ theory of covering surface and a type-function,we confirm the existence theorem of a Borel radius and a T-radius for the algebroidal function dealing with multiple values in the unit disc,which briefly extend some results for the algebroidal functions in the complex plane展开更多
In this paper,we investigate the growth relations between algebroid functions and their derivatives,and extend famous C.Chang inequality（see[1,4]）of meromorphic functions to algebroid functions.
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.
文摘A theorem of Maurer-Cartan type for Lie algebroids is presented. Suppose that any vector subbundle of a Lie algebroid is called interior differential system (IDS) for that Lie algebroid. A theorem of Frobenius type is obtained. Extending the classical notion of exterior diffential system (EDS) to Lie algebroids, a theorem of Cartan type is obtained.
基金supported by National Natural Science Foundation of China(Grant No.10871007)US-China CMR Noncommutative Geometry(Grant No.10911120391/A0109)+1 种基金China Postdoctoral Science Foundation(Grant No.20090451267)Science Research Foundation for Excellent Young Teachers of Mathematics School at Jilin University
文摘In this paper,we study the algebraic properties of the higher analogues of Courant algebroid structures on the direct sum bundle TM⊕∧nT*M for an m-dimensional manifold.As an application,we revisit Nambu-Poisson structures and multisymplectic structures.We prove that the graph of an(n+1)-vector fieldπis closed under the higher-order Dorfman bracket iffπis a Nambu-Poisson structure.Consequently,there is an induced Leibniz algebroid structure on∧nT*M.The graph of an(n+1)-formωis closed under the higher-order Dorfman bracket iffωis a premultisymplectic structure of order n,i.e.,dω=0.Furthermore,there is a Lie algebroid structure on the admissible bundle A∧nT*M.In particular,for a 2-plectic structure,it induces the Lie 2-algebra structure given in(Baez,Hoffnung and Rogers,2010).
基金supported by the National Natural Science Foundation of China(Grant Nos.11961049,11601219).
文摘In this paper,we study the structure of nonabelian omni-Lie algebroids.Firstly,taking Lie algebroid(E,[·,·]_(E,ρE))as the starting point,a nonabelian omni-Lie algebroid is defined on direct sum bundle DE⊕JE,where DE and JE are,respectively,the gauge Lie algebroid and the jet bundle of vector bundle E,and study its properties.Furthermore,it is concluded that the nonabelian omni-Lie algebroid is a trivial deformation of the omni-Lie algebroid,and the nonabelian omni-Lie algebroid is a matched pair of Leibniz algebroids.
基金supported by National Natural Science Foundation of China (Grant No. 11901221)。
文摘For any regular Lie algebroid A, the kernel K and the image F of its anchor map ρA, together with A itself fit into a short exact sequence, called the Atiyah sequence, of Lie algebroids. We prove that the Atiyah and Todd classes of dg manifolds arising from a regular Lie algebroid respect the Atiyah sequence, i.e.,the Atiyah and Todd classes of A restrict to the Atiyah and Todd classes of the bundle K of Lie algebras on the one hand, and project onto the Atiyah and Todd classes of the integrable distribution F■T_M on the other hand.
文摘Protobialgebroids include several kinds of algebroid structures such as Lie algebroid,Lie bialgebroid, Lie quasi-bialgebroid, etc. In this paper, the Dirac theories are generalized from Lie bialgebroid to protobialgebroid. We give the integrable conditions for a maximally isotropic subbundle being a Dirac structure for a protobialgebroid by the notion of a characteristic pair. From the integrable conditions, we found out that the Dirac structure has closed relations with the twisting of a protobialgebroid. At last, some special cases of the Dirac structures for protobialgebroids are discussed.
文摘Considering the prolongation of a Lie algebroid,the authors introduce Finsler algebroids and present important geometric objects on these spaces.Important endomorphisms like conservative and Barthel,Cartan tensor and some distinguished connections like Berwald,Cartan,Chern-Rund and Hashiguchi are introduced and studied.
基金supported by the National Natural Science Foundation of China(Grant No.19925105)the Research Project of"Nonlinear Science".
文摘For a transitive Lie algebroid A on a connected manifold M and its representation on a vector bundle F,we define a morphism of cohomology groups rk:Hk(A,F)→Hk(Lx,Fx),called the localization map,where Lx is the adjoint algebra at x∈M.The main result in this paper is that if M is simply connected,or H(LX,FX)is trivial,then T is injective.This means that the Lie algebroid 1-cohomology is totally determined by the 1-cohomology of its adjoint Lie algebra in the above two cases.
基金National Key Basic Research Project of China under Grant No.2004CB318000National Natural Science Foundation of China under Grant Nos.10375038 and 90403018
文摘After introducing some of the basic definitions and results from the theory of groupoid and Lie algebroid,we investigate the discrete Lagrangian mechanics from the viewpoint of groupoid theory and give the connection betweengroupoids variation and the methods of the first and second discrete variational principles.
文摘In this paper, the following theorem is obtained: Let w=w(z) be a υ valued algebroid function of finite order ρ>0 for |z|<∞ , then there exists a direction L: arg z=θ 0(0≤θ 0<2π) , such that for any δ>0(δ<π2 ), lim r→∞ log 3) (r,Δ(θ 0,δ),a) log r=ρ with 3υ-1 possible exceptions for any value of a . Where 3) (r,Δ(θ 0,δ),a) is the numbers of distinct zeros with multiplicity ≤3 of w(z)-a in (θ 0,δ)∩{||≤r} .
文摘Using the Nevanlinna theory of the value distribution of meromorphic functions, we investigate the existence problem of admissible algebroid solutions of generalized complex algebraic differential equations and obtain some results.
文摘In 1909 the brothers E. and F. Cosserat discovered a new nonlinear group theoretical approach to elasticity (EL), with the only experimental need to measure the EL constants. In a modern framework, they used the nonlinear Spencer sequence instead of the nonlinear Janet sequence for the Lie groupoid defining the group of rigid motions of space. Following H. Weyl, our purpose is to compute for the first time the linear and nonlinear Spencer sequences for the Lie groupoid defining the conformal group of space-time in order to provide the mathematical foundations of both electromagnetism (EM) and gravitation (GR), with the only experimental need to measure the EM and GR constants. With a manifold of dimension n ≥ 3, the difficulty is to deal with the n nonlinear transformations that have been called “elations” by E. Cartan in 1922. Using the fact that dimension n = 4 has very specific properties for the computation of the Spencer cohomology, we also prove that there is no conceptual difference between the (nonlinear) Cosserat EL field or induction equations and the (linear) Maxwell EM field or induction equations. As for gravitation, the dimension n = 4 also allows to have a conformal factor defined everywhere but at the central attractive mass because the inversion law of the isotropy subgroupoid made by second order jets transforms attraction into repulsion. The mathematical foundations of both electromagnetism and gravitation are thus only depending on the structure of the conformal pseudogroup of space-time.
文摘In 1909 the brothers E. and F. Cosserat discovered a new nonlinear group theoretical approach to elasticity (EL), with the only experimental need to measure the EL constants. In a modern language, their idea has been to use the nonlinear Spencer sequence instead of the nonlinear Janet sequence for the Lie groupoid defining the group of rigid motions of space. Following H. Weyl, our purpose is to compute for the first time the nonlinear Spencer sequence for the Lie groupoid defining the conformal group of space-time in order to provide the mathematical foundations of electromagnetism (EM), with the only experimental need to measure the EM constant in vacuum. With a manifold of dimension n, the difficulty is to deal with the n nonlinear transformations that have been called “elations” by E. Cartan in 1922. Using the fact that dimension n=4 has very specific properties for the computation of the Spencer cohomology, we prove that there is thus no conceptual difference between the Cosserat EL field or induction equations and the Maxwell EM field or induction equations. As a byproduct, the well known field/matter couplings (piezzoelectricity, photoelasticity, streaming birefringence, …) can be described abstractly, with the only experimental need to measure the corresponding coupling constants. The main consequence of this paper is the need to revisit the mathematical foundations of gauge theory (GT) because we have proved that EM was depending on the conformal group and not on U(1), with a shift by one step to the left in the physical interpretation of the differential sequence involved.
基金supported by NSFC (10871076,10771011)SRFDP (20050574002)NKBRP (2005CB321902)
文摘In this article, we first investigate the operational properties of algebroid functions. Then we prove two uniqueness theorems for algebroid functions.
基金The project Supported by NNSF of China(19971052)
文摘Using Nevanlinna theory of the value distribution of meromorphic functions, the author investigates the problem of the growth of solutions of two types of algebraic differential equation and obtains some results.
文摘In this paper, we investigate the normality relationship between algebroid multifunctions and their coefficient functions. We prove that the normality of a k-valued entire algebroid multifunctions family is equivalent to their coefficient functions in some conditions. Furthermore, we obtain some new normality criteria for algebroid multifunctions families based on these results. We also provide some examples to expound that some restricted conditions of our main results are necessary.
基金supported by NNSF of China(10471048)SRFDP(20050574002)
文摘In this article,the authors define the derived function of an algeboidal function in the unit disc,prove it is an algabriodal function,and study the order of algebroidal function and that of its derived function in unit circular disc.
基金Sponsored by the NSFC (10871076)the RFDP (20050574002)
文摘By using the definition of Hausdorff distance, we prove some normality criteria for families of meromorphic algebroid functions. Some examples are given to complement the theory in this article.
基金supported by the National Natural Science Foundation of China (11101096)
文摘Using Ahlfors’ theory of covering surface and a type-function,we confirm the existence theorem of a Borel radius and a T-radius for the algebroidal function dealing with multiple values in the unit disc,which briefly extend some results for the algebroidal functions in the complex plane
基金supported by National Natural Science Foundation of China(1047104810771011)the Fundamental Research Funds for the Central Universities
文摘In this paper,we investigate the growth relations between algebroid functions and their derivatives,and extend famous C.Chang inequality（see[1,4]）of meromorphic functions to algebroid functions.
