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The Yang—Baxter equation is reinvestigated in the framework of triple system. By requiring the rational R matrix of the Yang—Baxter equation satisfying the generalized Filippov condition, we derive a relation with respect to the rational R matrix. Moreover the case of the super Yang—Baxter equation is also investigated.

Based on the covariant prolongation structure technique, we construct the integrable higher-order deformations of the (2+1)-dimensional Heisenberg ferromagnet model and obtain their su(2) × R(λ) prolongation structures. By associating these deformed multidimensional Heisenberg ferromagnet models with the moving space curve in Euclidean space and using the Hasimoto function, we derive their geometrical equivalent counterparts, i.e., higher-order (2+1)-dimensional nonlinear Schrodinger equations.

In this paper, we investigate the categorical description of the boson oscillator. Based on the categories constructed by Khovanov, we introduce a categorification of the Fock states and the corresponding inner product of these states. We find that there are two different categorical definitions of the inner product of the Fock states. These two definitions are consistent with each other, and the decategorification results also coincide with those in conventional quantum mechanics. We also find that there are some interesting properties of the 2-morphisms which relate to the inner product of the states.

In this paper, we categorify the algebra Uq with the same approach as in [A. Lauda, Adv. Math. (2010), arXiv:math.QA/0803.3652; M. Khovanov, Comm. Algebra 11 (2001) 5033]. The algebra U = Uq is obtained from Uq by adjoining a collection of orthogonal idempotents 1λ, λ P, in which P is the weight lattice of Uq. Under such construction the algebra U is decomposed into a direct sum ⊕λP 1λ' U1λ. We set the collection of λ P as the objects of the category , 1-morphisms from λ to λ' are given by 1λ'U 1λ, and 2-morphisms are constructed by some semilinear form defined on U. Hence we get a 2-category from the algebra Uq.

We investigate some integrable modified Heisenberg ferromagnet models by using the prolongation structure theory. Through associating them with the motion of curve in Minkowski space, the corresponding coupled integrable equations are presented.

We propose the difference discrete variational principle in discrete mechanics and symplectic algorithm with variable step-length of time in finite duration based upon a noncommutative differential calculus established in this paper. This approach keeps both symplicticity and energy conservation discretely. We show that there exists the discrete version of the Euler-Lagrange cohomology in these discrete systems. We also discuss the solution existence in finite time-length and its site density in continuous limit, and apply our approach to the pendulum with periodic perturbation. The numerical results are satisfactory., 18 pages, 3 figures

In this letter, we present the general form of equations that generate a volume-preserving flow on a symplectic manifold (M, \omega). It is shown that every volume-preserving flow has some 2-forms acting the role of the Hamiltonian functions in the Hamiltonian mechanics and the ordinary Hamilton equations are included as a special case with a 2-form \frac{1}{n-1} H \omega where H is the corresponding Hamiltonian.

The Noether current and its variation relation with respect to diffeomorphism invariance of gravitational theories have been derived from the horizontal variation and vertical-horizontal bi-variation of the Lagrangian, respectively. For Einstein's GR in the stationary, axisymmetric black holes, the mass formula in vacuum can be derived from this Noether current although it definitely vanishes. This indicates that the mass formula of black holes is a vanishing Noether charge in this case. The first law of black hole thermodynamics can also be derived from the variation relation of this vanishing Noether current.

We introduce the Euler-Lagrange cohomology to study the symplectic and multisymplectic structures and their preserving properties in finite and infinite dimensional Lagrangian systems respectively We also explore their cei tain difference discrete counterparts in the relevant regularly discretized finite and infinite dimensional Lagrangian systems by means of the difference discrete variational principle with the difference being regarded as an entire geometric object and the noncommutative differential calculus on regular lattice. In order to show that in all these cases the symplectic and multisymplectic preserving properties do not necessarily depend on the relevant Euler-Lagrange equations, the Euler-Lagrange cohomological concepts and content in the configuration space are employed.

A new noncommutative differential calculus on function space of discrete Abelian groups is proposed. The derivatives are introduced with respect to the generators of the groups only. It is applied to discrete symplectic geometry and Hamiltonian systems with as well as the lattice gauge theory on regular lattice.

Based upon a fundamental principle, the generalized gauge principle, we construct a general model with G(L) x G(R)' x Z(2) gauge symmetry, where Z(2) = pi(4)(G(L)) is the fourth homotopy group of the gauge group G(L), by means of the non-commutative differential geometry and reformulating the standard model with the Higgs field being a gauge field on the fourth homotopy group of their gauge groups. We show that in this approach not only the Higgs held is automatically introduced on an equal footing with ordinary Yang-Mills gauge potentials and there are no extra constraints among the parameters at the tree level but also most importantly the models survive quantum corrections.

In this paper, we discuss the bicovariant differential calculus on quantum Lorentz group, and provide corresponding de Rham complex and Maurer-Cartan formulae.

In this note, we claim that we have obtained all nondegenerate solutions of Yang–Baxter equation with spectral parameter for square-lattice eight-vertex model without "zero-field" conditions by several solution transformations and six basic solutions with the help of Wu algorithm.

From the point that the Higgs fields can be regarded as the gauge fields on a discrete group, we construct a gauge theory of the left-right symmetric model SU(2)L × SU(2)R × U(1)B-L (LRSM) over Z2 ⊙ Z2 discrete symmetry.

In this paper, quantum subgroups and quantum co-set spaces including corresponding dififerential calculi are discussed.

In this paper, some new generalized inhomogeneous quantum groups corresponding to the homogeneous multiparameter quantum groups are constructed, furthermore, the -matrices for these inhomogeneous quantum groups are found.

We complete the construction of the gauge theory on discrete groups including fermions in the spirit of non-commutative geometry. We show that a simple Higgs field is such a gauge field with respect to -gauge symmetry over and the Yukawa coupling between Higgs and fermions is automatically introduced via minimum coupling principle. We also present an Abelian toy model.

Following the discussions on quantum groups by Reshetikhin, Takhadshyan and Faddeev, we thoroughly studied the differential calculus on quantum groups, and obtained the explicit expressione in the case of . We also compared our results with those of Woronowicz and Podles under the restriction of to .

We propose an alternative algebraic construction of the connection theory on the fiber bundle, and find its non-commutative extension, i.e., the connection theory based on guantum groups. In this construction, the Yang-Baxter equation arises as the exchange relations of connections and a deformed trace of quantum groups is introduced to build up the characteristic classes.

We introduce the Beltrami gauge symmetry related to the Beltrami parametrization of the metrics and the Beltrami equations. We explore its role in 2d induced gravity and show that Polyakov's-SL(2R) and KPZ's residual symmetries are subsymmetries of the Beltrami gauge symmetry in the light-cone gauge. We also find that the 2d induced gravity with or without matter can be reformulated as Beltrami-Liouville field theories on Riemann surfaces of higher genus.

The difference between the Riemann and Lorentz spinor manifolds of four dimensions is that the Dirac operator of the former is elliptic and that of the latter is hyperbolic. Moreover the spinor group of the former is a compact group and that of the latter is a noncompact group, which is isomorphic to SL(2,C). Hence the results and their interpretation coming from the two theories would be different. In this short note we study only the Lorentz spinor manifold and, especially, the solutions of Einstein-Dirac equations on the conformal space, which is closely related to the AdS/CFT correspondence.