We show that every source connected Lie groupoid always has global bisections through any given point. This bisection can be chosen to be the multiplication of some exponentials as close as possible to a prescribed cu...We show that every source connected Lie groupoid always has global bisections through any given point. This bisection can be chosen to be the multiplication of some exponentials as close as possible to a prescribed curve. The existence of bisections through more than one prescribed point is also discussed. We give some interesting applications of these results.展开更多
The notion of n-transitivity can be carried over from groups of diffeomorphisms on a manifold M to groups of bisections of a Lie groupoid over M. The main theorem states that the n-transitivity is fulfilled for all n ...The notion of n-transitivity can be carried over from groups of diffeomorphisms on a manifold M to groups of bisections of a Lie groupoid over M. The main theorem states that the n-transitivity is fulfilled for all n ∈N by an arbitrary group of Cr-bisections of a Lie groupoid F of class Cr, where 1 ≤ r ≤ ω, under mild conditions. For instance, the group of all bisections of any Lie groupoid and the group of all Lagrangian bisections of any symplectic groupoid are n-transitive in the sense of this theorem. In particular, if F is source connected for any arrow γ∈ Г, there is a bisection passing through γ.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 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 paper, we generalize the construction of the inverse transgression map done by Adem, A., Ruan, Y. and Zhang, B. in [A stringy product on twisted orbifold K-theory. Morfismos, 11, 33 64 （2007）] and give a dif...In this paper, we generalize the construction of the inverse transgression map done by Adem, A., Ruan, Y. and Zhang, B. in [A stringy product on twisted orbifold K-theory. Morfismos, 11, 33 64 （2007）] and give a different proof to the statement that the image of the inverse transgression map for a gerbe with connection over an orbifold is an inner local system on its inertia orbifold.展开更多
Let X denote a complex analytic manifold, and let Aut(X) denote the space of invertible maps of a germ (X, a) to a germ (X, b); this space is obviously a groupoid; roughly speaking, a 'Lie groupoid' is a subgr...Let X denote a complex analytic manifold, and let Aut(X) denote the space of invertible maps of a germ (X, a) to a germ (X, b); this space is obviously a groupoid; roughly speaking, a 'Lie groupoid' is a subgroupoid of Aut(X) defined by a system of partial differential equations.To a foliation with singularities on X one attaches such a groupoid, e.g. the smallest one whose Lie algebra contains the vector fields tangent to the foliation. It is called 'the Galois groupoid of the foliation'. Some examples are considered, for instance foliations of codimension one, and foliations defined by linear differential equations; in this last case one recuperates the usual differential Galois group.展开更多
文摘We show that every source connected Lie groupoid always has global bisections through any given point. This bisection can be chosen to be the multiplication of some exponentials as close as possible to a prescribed curve. The existence of bisections through more than one prescribed point is also discussed. We give some interesting applications of these results.
文摘The notion of n-transitivity can be carried over from groups of diffeomorphisms on a manifold M to groups of bisections of a Lie groupoid over M. The main theorem states that the n-transitivity is fulfilled for all n ∈N by an arbitrary group of Cr-bisections of a Lie groupoid F of class Cr, where 1 ≤ r ≤ ω, under mild conditions. For instance, the group of all bisections of any Lie groupoid and the group of all Lagrangian bisections of any symplectic groupoid are n-transitive in the sense of this theorem. In particular, if F is source connected for any arrow γ∈ Г, there is a bisection passing through γ.
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 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 National Natural Science Foundation of China(Grant No.11071176)
文摘In this paper, we generalize the construction of the inverse transgression map done by Adem, A., Ruan, Y. and Zhang, B. in [A stringy product on twisted orbifold K-theory. Morfismos, 11, 33 64 （2007）] and give a different proof to the statement that the image of the inverse transgression map for a gerbe with connection over an orbifold is an inner local system on its inertia orbifold.
文摘Let X denote a complex analytic manifold, and let Aut(X) denote the space of invertible maps of a germ (X, a) to a germ (X, b); this space is obviously a groupoid; roughly speaking, a 'Lie groupoid' is a subgroupoid of Aut(X) defined by a system of partial differential equations.To a foliation with singularities on X one attaches such a groupoid, e.g. the smallest one whose Lie algebra contains the vector fields tangent to the foliation. It is called 'the Galois groupoid of the foliation'. Some examples are considered, for instance foliations of codimension one, and foliations defined by linear differential equations; in this last case one recuperates the usual differential Galois group.
