Wang Shi-kun

Difference Discrete Variational Principle, Euler–Lagrange Cohomology and Symplectic, Multisymplectic Structures II: Euler–Lagrange Cohomology*

In this second paper of a series of papers, we explore the difference discrete versions for the Euler-Lagrange cohomology and apply them to the symplectic or multisymplectic geometry and their preserving properties in both the Lagrangian and Hamiltonian formalisms for discrete mechanics and field theory in the framework of multi-parameter differential approach. In terms of the difference discrete Euler-Lagrange cohomological concepts, we show that the symplectic or multisymplectic geometry and their difference discrete structure-preserving properties can always be established not only in the solution spaces of the discrete Euler-Lagrange or canonical equations derived by the difference discrete variational principle but also in the function space in each case if and only if the relevant closed Euler-Lagrange cohomological conditions are satisfied.

Difference Discrete Variational Principles, Euler–Lagrange Cohomology and Symplectic, Multisymplectic Structures I: Difference Discrete Variational Principle*

In this first paper of a series, we study the difference discrete variational principle in the framework of multi-parameter differential approach by regarding the forward difference as an entire geometric object in view of noncommutative differential geometry. Regarding the difference as an entire geometric object, the difference discrete version of Legendre transformation can be introduced. By virtue of this variational principle, we can discretely deal with the variation problems in both the Lagrangian and Hamiltonian formalisms to get difference discrete Euler-Lagrange equations and canonical ones for the difference discrete versions of the classical mechanics and classical field theory.

Difference Discrete Variational Principles, Euler?Lagrange Cohomology and Symplectic, Multisymplectic Structures III: Application to Symplectic and Multisymplectic Algorithms

In the previous papers I and II, we have studied the difference discrete variational principle and the Euler–Lagrange cohomology in the framework of multi-parameter differential approach. We have gotten the difference discrete Euler–Lagrange equations and canonical ones for the difference discrete versions of classical mechanics and field theory as well as the difference discrete versions for the Euler–Lagrange cohomology and applied them to get the necessary and sufficient condition for the symplectic or multisymplectic geometry preserving properties in both the Lagrangian and Hamiltonian formalisms. In this paper, we apply the difference discrete variational principle and Euler–Lagrange cohomological approach directly to the symplectic and multisymplectic algorithms. We will show that either Hamiltonian schemes or Lagrangian ones in both the symplectic and multisymplectic algorithms are variational integrators and their difference discrete symplectic structure-preserving properties can always be established not only in the solution space but also in the function space if and only if the related closed Euler–Lagrange cohomological conditions are satisfied.

Universal -matrix of the Super Yangian Double DY(gl(1|1))

Based on Drinfel ′ d realization of super Yangian Double DY (gl(1|1)), its pairing relations and universal R-matrix are given. By taking evaluation representation of universal R-matrix, another realization L ± (u) of DY (gl(1|1)) is obtained. These two realizations of DY (gl(1|1)) are related by the supersymmetric extension of Ding-Frenkel map. Yangian algebra was introduced by Drinfel ′ d[1, 2]. The quantum double of Yangian consists of Yangian itself and its dual with opposite comultiplication. There are three methods to define the Yangian and Yangian double: Drinfel ′ d-Jmbo [1, 3], Drinfel ′ d new realization [2] and RS approach [5] (or FRT approach [4] in the case of without center extension). The explicit isomorphism between Drinfel ′ d new realization and RS realization of Yangian double can be established through Gauss decomposition, the similar method used by Ding and Frenkel in the discussions of quantum Affine algebra. [6]. The property of Yangian double, such as quasi-triangular properties and equivalence of Drinfel ′ d and RS realization was studied well in some papers [7, 8, 9]. Although the Drinfel ′ d realization of super Yangian double [10, 11] was constructed by means of RS method and Gauss decomposition, the quasi-triangular property, such as universal R-matrix of super Yangian double has not been studied yet. In this paper, we find the Hopf pairing relations between super Yangian Y (gl(1|1)) and its dual, then construct the universal R-matrix of DY (gl(1|1)). By taking evaluation representation, we get the FRT realization of DY (gl(1|1)). Super Yangian double DY (gl(1|1)) is the Hopf algebra generated by elementsBased on Drinfeld realization of super Yangian double DY(gl(1|1)), its pairing relations and universal -matrix are given. By taking evaluation representation of universal -matrix, another realization L±(u) of DY(gl(1|1)) is obtained. These two realizations of DY(gl(1|1)) are related by the supersymmetric extension of Ding-fienkel map.

Anomalies, Cohomology and Chiern-Simons Cochains

We present a concept of cohomology of the gauge transformations, establish the homomorphic relation between this cohomology group and the Chern-Simons type characteristic class sequence, and analyse their applications to the non-Abelian anomaly, the gauge invariant Wess-Zumino effective action and the Schwinger term etc. We show that the present approach includes faddeevs songs and zuminos approaches.

CHERN-SIMONS TYPE CHARACTERISTIC CLASSES

It is shown that one can generalize the concept of well-known Chern-Simons secondary characteristic classes to introducing a sequence of new characteristic classes named the Chern-Simons type characteristic classes. One of the important properties is given by a theorem which can be interpreted as that the exterior differential of a th Chern-Simons type characteristic class is exactly equal to the coboundary of the cochain of the -1st Chern-Simons type characteristic classes.

A free-fermion type solution of quantum dynamical Yang-Baxter equation

The symmetries of the quantum dynamical Yang-Baxter (QDYB) equation without spectral parameters for gl(2) are discussed. The classification of the six-vertex type solutions to the QDYB equation without spectral parameters is given and a free-fermion type solution is obtained.

Prolongation Structures of Nonlinear Systems in Higher Dimensions

It is shown that the prolongation structure theory for nonlinear (evolution) equations with two independent variables can be generalized to the systems with many independent variables. By means of the nonlinear realization theory of gauge symmetries, the fundamental equations for prolongation structures and the requirements for the generalized Lax representations of the nonlinear systems in higher dimensions have been given. Based upon the invariances of the prolongation structures or the generalized Lax representation under certain transformations, the general condition satisfied by the auto-Backlund transformations has been proposed and searching for a kind of auto-Backlund transformations has been transferred to solving the regular Riemann-Hilbert problem.

DIFFEOMORPHISM AND BELTRAMI ALGEBRAS ON TORUS

The infinite-dimensional algebras on two-dimensional torus are developed. In particular the diffeomorphism algebra and the Beltrami algebra on torus with their central extensions have been presented. It is shown that these algebras are corresponding to the diffeomorphism and the quasiconformal transformation of torus, respectively.

Principal Riemann-Hilbert problem and N-fold charged Kerr solution

The authors introduced the principal Riemann-Hilbert problem (PRHP) (1983) to solve the Einstein electrovac field equations and emphasised the group properties of the PRHP, whereupon they found the N-fold charged Kerr solution which may be considered to be an electrovac generalisation of Neugebauers N-fold Kerr solution (1980). In this paper, they show the topics systematically with some details.