Denote a finite dimensional Hopf C*-algebra by H, and a Hopf *-subalgebra of H by H1. In this paper, we study the construction of the field algebra in Hopf spin models determined by H1 together with its symmetry. More...Denote a finite dimensional Hopf C*-algebra by H, and a Hopf *-subalgebra of H by H1. In this paper, we study the construction of the field algebra in Hopf spin models determined by H1 together with its symmetry. More precisely, we consider the quantum double D(H, H_(1)) as the bicrossed product of the opposite dual Hopˆ of H and H1 with respect to the coadjoint representation, the latter acting on the former and vice versa, and under the non-trivial commutation relations between H1 and Ĥ we define the observable algebra AH1. Then using a comodule action of D(H, H1) on AH1, we obtain the field algebra FH1, which is the crossed product AH1⋊D(H,H_(1)), and show that the observable algebra AH1 is exactly a D(H, H1)-invariant subalgebra of FH1. Furthermore, we prove that there exists a duality between D(H, H1) and AH1, implemented by a*-homomorphism of D(H, H_(1)).展开更多
Let F be the field algebra of G -spin model, D(G) the double algebra of a finite group G and D(H) the sub-Hopf algerba of D(G) determined by the subgroup H of G . The paper builds a correspondence between D(H) and th...Let F be the field algebra of G -spin model, D(G) the double algebra of a finite group G and D(H) the sub-Hopf algerba of D(G) determined by the subgroup H of G . The paper builds a correspondence between D(H) and the D(H) -invariant sub- C * -algebra A H in F, and proves that the correspondence is strictly monotonic.展开更多
Field algebra of G-spin models can provide the simplest examples of lattice field theory exhibiting quantum symmetry. Let D(G) be the double algebra of a finite group G and D(H), a sub-algebra of D(G) determined by s...Field algebra of G-spin models can provide the simplest examples of lattice field theory exhibiting quantum symmetry. Let D(G) be the double algebra of a finite group G and D(H), a sub-algebra of D(G) determined by subgroup H of G. This paper gives concrete generators and the structure of the observable algebra A H, which is a D(H)-invariant sub-algebra in the field algebra of G-spin models F, and shows that A H is a C *-algebra. The correspondence between H and A H is strictly monotonic. Finally, a duality between D(H) and A H is given via an irreducible vacuum C *-representation of F.展开更多
In this article, we revisit some aspects of the computation of the cohomology class of H2 (Witt, C)?using some methods in two-dimensional conformal field theory and conformal algebra to obtain the one-dimensional cent...In this article, we revisit some aspects of the computation of the cohomology class of H2 (Witt, C)?using some methods in two-dimensional conformal field theory and conformal algebra to obtain the one-dimensional central extension of the Witt algebra to the Virasoro algebra. Even though this is well-known in the context of standard mathematical physics literature, the operator product expansion of the energy-momentum tensor in two-dimensional conformal field theory is presented almost axiomatically. In this paper, we attempt to reformulate it with the help of a suitable modification of conformal algebra (as developed by V. Kac), and apply it to compute the representative element of the cohomology class which gives the desired central extension. This paper was written in the scope of an undergraduate’s exploration of conformal field theory and to gain insight on the subject from a mathematical perspective.展开更多
This paper derives explicit expressions of the infinitesimal gauge operators for pseudoscalalr fields in a gauge theory coupling vector and axial-vector fields with the aid of the method of operator algebra. The gauge...This paper derives explicit expressions of the infinitesimal gauge operators for pseudoscalalr fields in a gauge theory coupling vector and axial-vector fields with the aid of the method of operator algebra. The gauge operators of the coset pure gauge fields theory under the chiral group SU(N) X SU(N) are also obtainede.展开更多
Finite fields form an important chapter in abstract algebra, and mathematics in general, yet the traditional expositions, part of Abstract Algebra courses, focus on the axiomatic presentation, while Ramification Theor...Finite fields form an important chapter in abstract algebra, and mathematics in general, yet the traditional expositions, part of Abstract Algebra courses, focus on the axiomatic presentation, while Ramification Theory in Algebraic Number Theory, making a suited topic for their applications, is usually a separated course. We aim to provide a geometric and intuitive model for finite fields, involving algebraic numbers, in order to make them accessible and interesting to a much larger audience, and bridging the above mentioned gap. Such lattice models of finite fields provide a good basis for later developing their study in a more concrete way, including decomposition of primes in number fields, Frobenius elements, and Frobenius lifts, allowing to approach more advanced topics, such as Artin reciprocity law and Weil Conjectures, while keeping the exposition to the concrete level of familiar number systems. Examples are provided, intended for an undergraduate audience in the first place.展开更多
This paper unfolds and reviews the theory of abstract algebra, field extensions and discusses various kinds of field extensions. Field extensions are said to be algebraic or transcendental. We pay much attention to al...This paper unfolds and reviews the theory of abstract algebra, field extensions and discusses various kinds of field extensions. Field extensions are said to be algebraic or transcendental. We pay much attention to algebraic extensions. Finally, we construct finite extensions of Q and finite extensions of the function field over finite field F<sub>p </sub>using the notion of field completion, analogous to field extensions. With the study of field extensions, considering any polynomial with coefficients in the field, we can find the roots of the polynomial, and with the notion of algebraically closed fields, we have one field, F, where we can find the roots of any polynomial with coefficients in F.展开更多
Multidimensional noncommutative Laplace transforms over octonions are studied. Theorems about direct and inverse transforms and other properties of the Laplace transforms over the Cayley-Dickson algebras are proved. A...Multidimensional noncommutative Laplace transforms over octonions are studied. Theorems about direct and inverse transforms and other properties of the Laplace transforms over the Cayley-Dickson algebras are proved. Applications to partial differential equations including that of elliptic, parabolic and hyperbolic type are investigated. Moreover, partial differential equations of higher order with real and complex coefficients and with variable coefficients with or without boundary conditions are considered.展开更多
Theoretically, it is plausible to assume for a chosen charge distribution the electric field can be calculated. However, in practice depending on the geometry of the distribution one faces mathematical challenges. In ...Theoretically, it is plausible to assume for a chosen charge distribution the electric field can be calculated. However, in practice depending on the geometry of the distribution one faces mathematical challenges. In this research- oriented project, we select a set of related familiar 2D geometric curves addressing the mathematical issues. Specifically, we consider a family of curves that evolved via step-by-step “evolution”. The evolution begins from a segment of a circular arc to a complete circle. The electric fields are formulated, evaluated, and graphed. Accomplishing these objectives relied heavily on utilizing a Computer Algebra System (CAS), specifically Mathematica [1]. The CPU’s expensive runtimes are circumvented by introducing mathematical procedures.展开更多
The electric field of a 3D spherical uniform charge distribution embodying a spherical mobile void at an exterior point is calculated. The size of the void and its path is arbitrary. Specifically, three different traj...The electric field of a 3D spherical uniform charge distribution embodying a spherical mobile void at an exterior point is calculated. The size of the void and its path is arbitrary. Specifically, three different trajectories are analyzed. The movement of the void impacts the electric field so that the field becomes time-dependent. In terms of the chosen path and the size of the bubble, we evaluate the time-dependent electric field. The time profile of the field is calculated. Because of the computational challenges, the most of calculation is carried out utilizing a Computer Algebra System (CAS), specifically Mathematica [1]. This project makes the CAS an essential tool not only for calculating the field but for animating the features of the mobile void. An atlas of the study cases is included.展开更多
As a general feature, the electric field of a localized electric charge distribution diminishes as the distance from the distribution increases;there are exceptions to this feature. For instance, the electric field of...As a general feature, the electric field of a localized electric charge distribution diminishes as the distance from the distribution increases;there are exceptions to this feature. For instance, the electric field of a charged ring (being a localized charge distribution) along its symmetry axis perpendicular to the ring through its center rather than as expected being a diminishing field encounters a local maximum bump. It is the objective of this research-oriented study to analyze the impact of this bump on the characteristics of a massive point-like charged particle oscillating along the symmetry axis. Two scenarios with and without gravity along the symmetry axis are considered. In addition to standard kinematic diagrams, various phase diagrams conducive to a better understanding are constructed. Applying Computer Algebra System (CAS), [1] [2] most calculations are carried out symbolically. Finally, by assigning a set of reasonable numeric parameters to the symbolic quantities various 3D animations are crafted. All the CAS codes are included.展开更多
基金supported by National Nature Science Foundation of China(11871303,11701423)Nature Science Foundation of Hebei Province(A2019404009)。
文摘Denote a finite dimensional Hopf C*-algebra by H, and a Hopf *-subalgebra of H by H1. In this paper, we study the construction of the field algebra in Hopf spin models determined by H1 together with its symmetry. More precisely, we consider the quantum double D(H, H_(1)) as the bicrossed product of the opposite dual Hopˆ of H and H1 with respect to the coadjoint representation, the latter acting on the former and vice versa, and under the non-trivial commutation relations between H1 and Ĥ we define the observable algebra AH1. Then using a comodule action of D(H, H1) on AH1, we obtain the field algebra FH1, which is the crossed product AH1⋊D(H,H_(1)), and show that the observable algebra AH1 is exactly a D(H, H1)-invariant subalgebra of FH1. Furthermore, we prove that there exists a duality between D(H, H1) and AH1, implemented by a*-homomorphism of D(H, H_(1)).
文摘Let F be the field algebra of G -spin model, D(G) the double algebra of a finite group G and D(H) the sub-Hopf algerba of D(G) determined by the subgroup H of G . The paper builds a correspondence between D(H) and the D(H) -invariant sub- C * -algebra A H in F, and proves that the correspondence is strictly monotonic.
基金Supported by the National Natural Science Foundationof China (No.10 0 0 10 2 0 )
文摘Field algebra of G-spin models can provide the simplest examples of lattice field theory exhibiting quantum symmetry. Let D(G) be the double algebra of a finite group G and D(H), a sub-algebra of D(G) determined by subgroup H of G. This paper gives concrete generators and the structure of the observable algebra A H, which is a D(H)-invariant sub-algebra in the field algebra of G-spin models F, and shows that A H is a C *-algebra. The correspondence between H and A H is strictly monotonic. Finally, a duality between D(H) and A H is given via an irreducible vacuum C *-representation of F.
文摘In this article, we revisit some aspects of the computation of the cohomology class of H2 (Witt, C)?using some methods in two-dimensional conformal field theory and conformal algebra to obtain the one-dimensional central extension of the Witt algebra to the Virasoro algebra. Even though this is well-known in the context of standard mathematical physics literature, the operator product expansion of the energy-momentum tensor in two-dimensional conformal field theory is presented almost axiomatically. In this paper, we attempt to reformulate it with the help of a suitable modification of conformal algebra (as developed by V. Kac), and apply it to compute the representative element of the cohomology class which gives the desired central extension. This paper was written in the scope of an undergraduate’s exploration of conformal field theory and to gain insight on the subject from a mathematical perspective.
文摘This paper derives explicit expressions of the infinitesimal gauge operators for pseudoscalalr fields in a gauge theory coupling vector and axial-vector fields with the aid of the method of operator algebra. The gauge operators of the coset pure gauge fields theory under the chiral group SU(N) X SU(N) are also obtainede.
文摘Finite fields form an important chapter in abstract algebra, and mathematics in general, yet the traditional expositions, part of Abstract Algebra courses, focus on the axiomatic presentation, while Ramification Theory in Algebraic Number Theory, making a suited topic for their applications, is usually a separated course. We aim to provide a geometric and intuitive model for finite fields, involving algebraic numbers, in order to make them accessible and interesting to a much larger audience, and bridging the above mentioned gap. Such lattice models of finite fields provide a good basis for later developing their study in a more concrete way, including decomposition of primes in number fields, Frobenius elements, and Frobenius lifts, allowing to approach more advanced topics, such as Artin reciprocity law and Weil Conjectures, while keeping the exposition to the concrete level of familiar number systems. Examples are provided, intended for an undergraduate audience in the first place.
文摘This paper unfolds and reviews the theory of abstract algebra, field extensions and discusses various kinds of field extensions. Field extensions are said to be algebraic or transcendental. We pay much attention to algebraic extensions. Finally, we construct finite extensions of Q and finite extensions of the function field over finite field F<sub>p </sub>using the notion of field completion, analogous to field extensions. With the study of field extensions, considering any polynomial with coefficients in the field, we can find the roots of the polynomial, and with the notion of algebraically closed fields, we have one field, F, where we can find the roots of any polynomial with coefficients in F.
文摘Multidimensional noncommutative Laplace transforms over octonions are studied. Theorems about direct and inverse transforms and other properties of the Laplace transforms over the Cayley-Dickson algebras are proved. Applications to partial differential equations including that of elliptic, parabolic and hyperbolic type are investigated. Moreover, partial differential equations of higher order with real and complex coefficients and with variable coefficients with or without boundary conditions are considered.
文摘Theoretically, it is plausible to assume for a chosen charge distribution the electric field can be calculated. However, in practice depending on the geometry of the distribution one faces mathematical challenges. In this research- oriented project, we select a set of related familiar 2D geometric curves addressing the mathematical issues. Specifically, we consider a family of curves that evolved via step-by-step “evolution”. The evolution begins from a segment of a circular arc to a complete circle. The electric fields are formulated, evaluated, and graphed. Accomplishing these objectives relied heavily on utilizing a Computer Algebra System (CAS), specifically Mathematica [1]. The CPU’s expensive runtimes are circumvented by introducing mathematical procedures.
文摘The electric field of a 3D spherical uniform charge distribution embodying a spherical mobile void at an exterior point is calculated. The size of the void and its path is arbitrary. Specifically, three different trajectories are analyzed. The movement of the void impacts the electric field so that the field becomes time-dependent. In terms of the chosen path and the size of the bubble, we evaluate the time-dependent electric field. The time profile of the field is calculated. Because of the computational challenges, the most of calculation is carried out utilizing a Computer Algebra System (CAS), specifically Mathematica [1]. This project makes the CAS an essential tool not only for calculating the field but for animating the features of the mobile void. An atlas of the study cases is included.
文摘As a general feature, the electric field of a localized electric charge distribution diminishes as the distance from the distribution increases;there are exceptions to this feature. For instance, the electric field of a charged ring (being a localized charge distribution) along its symmetry axis perpendicular to the ring through its center rather than as expected being a diminishing field encounters a local maximum bump. It is the objective of this research-oriented study to analyze the impact of this bump on the characteristics of a massive point-like charged particle oscillating along the symmetry axis. Two scenarios with and without gravity along the symmetry axis are considered. In addition to standard kinematic diagrams, various phase diagrams conducive to a better understanding are constructed. Applying Computer Algebra System (CAS), [1] [2] most calculations are carried out symbolically. Finally, by assigning a set of reasonable numeric parameters to the symbolic quantities various 3D animations are crafted. All the CAS codes are included.
