The purpose of this paper is to present for the first time an elementary summary of a few recent results obtained through the application of the formal theory of partial differential equations and Lie pseudogroups in ...The purpose of this paper is to present for the first time an elementary summary of a few recent results obtained through the application of the formal theory of partial differential equations and Lie pseudogroups in order to revisit the mathematical foundations of general relativity. Other engineering examples (control theory, elasticity theory, electromagnetism) will also be considered in order to illustrate the three fundamental results that we shall provide successively. 1) VESSIOT VERSUS CARTAN: The quadratic terms appearing in the “Riemann tensor” according to the “Vessiot structure equations” must not be identified with the quadratic terms appearing in the well known “Cartan structure equations” for Lie groups. In particular, “curvature + torsion” (Cartan) must not be considered as a generalization of “curvature alone” (Vessiot). 2) JANET VERSUS SPENCER: The “Ricci tensor” only depends on the nonlinear transformations (called “elations” by Cartan in 1922) that describe the “difference” existing between the Weyl group (10 parameters of the Poincaré subgroup + 1 dilatation) and the conformal group of space-time (15 parameters). It can be defined without using the indices leading to the standard contraction or trace of the Riemann tensor. Meanwhile, we shall obtain the number of components of the Riemann and Weyl tensors without any combinatoric argument on the exchange of indices. Accordingly and contrary to the “Janet sequence”, the “Spencer sequence” for the conformal Killing system and its formal adjoint fully describe the Cosserat equations, Maxwell equations and Weyl equations but General Relativity is not coherent with this result. 3) ALGEBRA VERSUS GEOMETRY: Using the powerful methods of “Algebraic Analysis”, that is a mixture of homological agebra and differential geometry, we shall prove that, contrary to other equations of physics (Cauchy equations, Cosserat equations, Maxwell equations), the Einstein equations cannot be “parametrized”, that is the g展开更多
In recent papers, a few physicists studying Black Hole perturbation theory in General Relativity (GR) have tried to construct the initial part of a differential sequence based on the Kerr metric, using methods similar...In recent papers, a few physicists studying Black Hole perturbation theory in General Relativity (GR) have tried to construct the initial part of a differential sequence based on the Kerr metric, using methods similar to the ones they already used for studying the Schwarzschild geometry. Of course, such a differential sequence is well known for the Minkowski metric and successively contains the Killing (order 1), the Riemann (order 2) and the Bianchi (order 1 again) operators in the linearized framework, as a particular case of the Vessiot structure equations. In all these cases, they discovered that the compatibility conditions (CC) for the corresponding Killing operator were involving a mixture of both second order and third order CC and their idea has been to exhibit only a minimal number of generating ones. Unhappily, these physicists are neither familiar with the formal theory of systems of partial differential equations and differential modules, nor with the formal theory of Lie pseudogroups. Hence, even if they discovered a link between these differential sequences and the number of parameters of the Lie group preserving the background metric, they have been unable to provide an intrinsic explanation of this fact, being limited by the technical use of Weyl spinors, complex Teukolsky scalars or Killing-Yano tensors. The purpose of this difficult computational paper is to provide differential and homological methods in order to revisit and solve these questions, not only in the previous cases but also in the specific case of any Lie group or Lie pseudogroup of transformations. These new tools, which are now available as computer algebra packages, question the mathematical foundations of GR and the origin of gravitational waves.展开更多
We consider whether the tilting properties of a tilting A-module T and a tilting B-module T′ can convey to their tensor product T T′. The main result is that T T′ turns out to be an （n ＋ m）-tilting A B-modu...We consider whether the tilting properties of a tilting A-module T and a tilting B-module T′ can convey to their tensor product T T′. The main result is that T T′ turns out to be an （n ＋ m）-tilting A B-module, where T is an m-tilting A-module and T′ is an n-tilting B-module.展开更多
We present a general construction producing a unitary corepresentation of a multiplier Hopf algebra in itself and study RR-corepresentations of twisted tensor coproduct multiplier Hopf algebra. Then we investigate som...We present a general construction producing a unitary corepresentation of a multiplier Hopf algebra in itself and study RR-corepresentations of twisted tensor coproduct multiplier Hopf algebra. Then we investigate some properties of functors, integrals and morphisms related to the categories of the crossed modules and covariant modules over multiplier Hopf algebras. By doing so, we can apply our theory to the case of group-cograded multiplier Hopf algebras, in particular, the case of Hopf group-coalgebras.展开更多
In this paper,we study irreducible non-weight modules over the mirror Heisenberg-Virasoro algebra D,including Whittaker modules,U(Cd_(0))-free modules and their tensor products.More precisely,we give the necessary and...In this paper,we study irreducible non-weight modules over the mirror Heisenberg-Virasoro algebra D,including Whittaker modules,U(Cd_(0))-free modules and their tensor products.More precisely,we give the necessary and sufficient conditions for the Whittaker modules to be irreducible.We determine all the D-module structures on U(Cd_(0)),and find the necessary and sufficient conditions for these modules to be irreducible.At last,we determine the necessary and sufficient conditions for the tensor products of Whittaker modules and U(Cd_(0))-free modules to be irreducible,and obtain that any two such tensor products are isomorphic if and only if the corresponding Whittaker modules and U(Cd_(0))-free modules are isomorphic.These lead to many new irreducible non-weight modules over D.展开更多
In this paper,we study the tensor module P■M over the Witt superalgebra W_(m,n)~+(resp.W_(m,n)),where P is a simple module over the Weyl superalgebra K_(m,n)~+(resp.K_(m,n))and M is a simple weight module over the ge...In this paper,we study the tensor module P■M over the Witt superalgebra W_(m,n)~+(resp.W_(m,n)),where P is a simple module over the Weyl superalgebra K_(m,n)~+(resp.K_(m,n))and M is a simple weight module over the general linear Lie superalgebra gl(m,n).We obtain the necessary and sufficient conditions for P■M to be simple,and determine all the simple subquotients of P■M when it is not simple.All the work leads to the completion of some classification problems on the weight representation theories of W_(m,n)~+and W_(m,n).展开更多
It is known that the Schrddinger-Virasoro algebras, including the original Schrddinger-Virasoro algebra and the twisted Schr?dinger-Virasoro algebra, are playing important roles in mathematics and statistical physics....It is known that the Schrddinger-Virasoro algebras, including the original Schrddinger-Virasoro algebra and the twisted Schr?dinger-Virasoro algebra, are playing important roles in mathematics and statistical physics. In this paper, we study the tensor products of weight modules over the Schr?dinger- Virasoro algebras. The irreducibility criterion for the tensor products of highest weight modules with intermediate series modules over the Schr?dinger-Virasoro algebra is obtained.展开更多
A classical question of Yoneda asks when the tensor product of two injective modules is injective. A complete answer to this question was given by Enochs and Jenda in 1991. In this paper the analogue question for pure...A classical question of Yoneda asks when the tensor product of two injective modules is injective. A complete answer to this question was given by Enochs and Jenda in 1991. In this paper the analogue question for pure-injective modules is studied.展开更多
文摘The purpose of this paper is to present for the first time an elementary summary of a few recent results obtained through the application of the formal theory of partial differential equations and Lie pseudogroups in order to revisit the mathematical foundations of general relativity. Other engineering examples (control theory, elasticity theory, electromagnetism) will also be considered in order to illustrate the three fundamental results that we shall provide successively. 1) VESSIOT VERSUS CARTAN: The quadratic terms appearing in the “Riemann tensor” according to the “Vessiot structure equations” must not be identified with the quadratic terms appearing in the well known “Cartan structure equations” for Lie groups. In particular, “curvature + torsion” (Cartan) must not be considered as a generalization of “curvature alone” (Vessiot). 2) JANET VERSUS SPENCER: The “Ricci tensor” only depends on the nonlinear transformations (called “elations” by Cartan in 1922) that describe the “difference” existing between the Weyl group (10 parameters of the Poincaré subgroup + 1 dilatation) and the conformal group of space-time (15 parameters). It can be defined without using the indices leading to the standard contraction or trace of the Riemann tensor. Meanwhile, we shall obtain the number of components of the Riemann and Weyl tensors without any combinatoric argument on the exchange of indices. Accordingly and contrary to the “Janet sequence”, the “Spencer sequence” for the conformal Killing system and its formal adjoint fully describe the Cosserat equations, Maxwell equations and Weyl equations but General Relativity is not coherent with this result. 3) ALGEBRA VERSUS GEOMETRY: Using the powerful methods of “Algebraic Analysis”, that is a mixture of homological agebra and differential geometry, we shall prove that, contrary to other equations of physics (Cauchy equations, Cosserat equations, Maxwell equations), the Einstein equations cannot be “parametrized”, that is the g
文摘In recent papers, a few physicists studying Black Hole perturbation theory in General Relativity (GR) have tried to construct the initial part of a differential sequence based on the Kerr metric, using methods similar to the ones they already used for studying the Schwarzschild geometry. Of course, such a differential sequence is well known for the Minkowski metric and successively contains the Killing (order 1), the Riemann (order 2) and the Bianchi (order 1 again) operators in the linearized framework, as a particular case of the Vessiot structure equations. In all these cases, they discovered that the compatibility conditions (CC) for the corresponding Killing operator were involving a mixture of both second order and third order CC and their idea has been to exhibit only a minimal number of generating ones. Unhappily, these physicists are neither familiar with the formal theory of systems of partial differential equations and differential modules, nor with the formal theory of Lie pseudogroups. Hence, even if they discovered a link between these differential sequences and the number of parameters of the Lie group preserving the background metric, they have been unable to provide an intrinsic explanation of this fact, being limited by the technical use of Weyl spinors, complex Teukolsky scalars or Killing-Yano tensors. The purpose of this difficult computational paper is to provide differential and homological methods in order to revisit and solve these questions, not only in the previous cases but also in the specific case of any Lie group or Lie pseudogroup of transformations. These new tools, which are now available as computer algebra packages, question the mathematical foundations of GR and the origin of gravitational waves.
基金Acknowledgements This work was supported by the National Natural Science Foundation of China （Grant Nos. 11471269, 61373140）, the Natural Science Foundation of Fujian Province （2016J01002）, and 2016 Incubation Program for Scientific Research Talent of Distinguished Young of Colledges and Universities in Fujian Province.
文摘We consider whether the tilting properties of a tilting A-module T and a tilting B-module T′ can convey to their tensor product T T′. The main result is that T T′ turns out to be an （n ＋ m）-tilting A B-module, where T is an m-tilting A-module and T′ is an n-tilting B-module.
基金Supported by National Natural Science Foundation of China (Grant No. 10871042) the Research Fund for the Doctoral Program of Higher Education (Grant No. 20060286006)+1 种基金 Natural Science Foundation of Jiangsu Province (Grant No. BK2009258)the Key Project of Chinese Ministry of Education of China (Grant No.108154)
文摘We present a general construction producing a unitary corepresentation of a multiplier Hopf algebra in itself and study RR-corepresentations of twisted tensor coproduct multiplier Hopf algebra. Then we investigate some properties of functors, integrals and morphisms related to the categories of the crossed modules and covariant modules over multiplier Hopf algebras. By doing so, we can apply our theory to the case of group-cograded multiplier Hopf algebras, in particular, the case of Hopf group-coalgebras.
基金supported by China Scholarship Council(Grant No.201906340096)National Natural Science Foundation of China(Grant Nos.11771410 and 11931009)+2 种基金supported by National Natural Science Foundation of China(Grant No.11801066)supported by National Natural Science Foundation of China(Grant No.11871190)Natural Sciences and Engineering Research Council of Canada(Grant No.311907-2020).
文摘In this paper,we study irreducible non-weight modules over the mirror Heisenberg-Virasoro algebra D,including Whittaker modules,U(Cd_(0))-free modules and their tensor products.More precisely,we give the necessary and sufficient conditions for the Whittaker modules to be irreducible.We determine all the D-module structures on U(Cd_(0)),and find the necessary and sufficient conditions for these modules to be irreducible.At last,we determine the necessary and sufficient conditions for the tensor products of Whittaker modules and U(Cd_(0))-free modules to be irreducible,and obtain that any two such tensor products are isomorphic if and only if the corresponding Whittaker modules and U(Cd_(0))-free modules are isomorphic.These lead to many new irreducible non-weight modules over D.
基金supported by National Natural Science Foundation of China(Grant Nos.11971440,11801390 and 11871052)。
文摘In this paper,we study the tensor module P■M over the Witt superalgebra W_(m,n)~+(resp.W_(m,n)),where P is a simple module over the Weyl superalgebra K_(m,n)~+(resp.K_(m,n))and M is a simple weight module over the general linear Lie superalgebra gl(m,n).We obtain the necessary and sufficient conditions for P■M to be simple,and determine all the simple subquotients of P■M when it is not simple.All the work leads to the completion of some classification problems on the weight representation theories of W_(m,n)~+and W_(m,n).
基金the National Natural Science Foundation of China(Grant Nos.11571145,11871249)the Natural Science Foundation of Zhejiang Province(No.LZ14A010001).
文摘It is known that the Schrddinger-Virasoro algebras, including the original Schrddinger-Virasoro algebra and the twisted Schr?dinger-Virasoro algebra, are playing important roles in mathematics and statistical physics. In this paper, we study the tensor products of weight modules over the Schr?dinger- Virasoro algebras. The irreducibility criterion for the tensor products of highest weight modules with intermediate series modules over the Schr?dinger-Virasoro algebra is obtained.
文摘A classical question of Yoneda asks when the tensor product of two injective modules is injective. A complete answer to this question was given by Enochs and Jenda in 1991. In this paper the analogue question for pure-injective modules is studied.
