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In this paper, we lift Fermions to functors acting on some homotopy category by the Boson-Fermion correspondence and get the categorified relations of Fermions. In this way, both the categorified Bosons and the categorified Fermions can be viewed as functors on the same category. We also give actions of these functors on the charged Young diagrams (or equivalently Maya diagrams), so that the classical theory of Boson-Fermion correspondence is very well recovered as a result of such a categorification.

We construct a fermion analogue of the Fock representation of quantum toroidal algebra and construct the fermion representation of quantum toroidal algebra on the K-theory of Hilbert scheme.

We categorify the notion of coalgebras by imposing a co-associative law up to some isomorphisms on the co-multiplication map and requiring that these isomorphisms satisfy certain law of their own, which is called the co-pentagon identity. We also set up a 2-category of 2-coalgebras. The purpose of this study is from the idea of reconsidering the quasi-Hopf algebras by the categorification process, so that we can study the theory of quasi-Hopf algebras and their representations in some new framework of higher category theory in natural ways.

We construct the integrable deformations of the Heisenberg supermagnet model with the quadratic constraints (i) S2 = 3S − 2I, for S USPL(2/1)/S(U(2) × U(1)) and (ii) S2 = S, for S USPL(2/1)/S(L(1/1) × U(1)). Under the gauge transformation, their corresponding gauge equivalent counterparts are derived. They are the Grassman odd and super mixed derivative nonlinear Schrodinger equation, respectively.

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 between groupoids variation and the methods of the first and second discrete variational principles.

We present the noncommutative differential calculus on the function space of the infinite set and construct a homotopy operator to prove the analogue of the Poincare lemma for the difference complex. Then the horizontal and vertical complexes are introduced with the total differential map and vertical exterior derivative. As the application of the differential calculus, we derive the schemes with the conservation of symplecticity and energy for Hamiltonian system and a two-dimensional integral models with infinite sequence of conserved currents. Then an Euler-Lagrange cohomology with symplectic structure-preserving is given in the discrete classical mechanics.

Based on noncommutative differential calculus, we present a theory of prolongation structure for semidiscrete nonlinear evolution equations. As an illustrative example, a semi-discrete model of the nonlinear Schrodinger equation is discussed in terms of this theory and the corresponding Lax pairs are also given.

The representation (p,q) of spin group Spin(p,q) in any dimensional space is given by induction, and the relation between two representations, which are obtained in two kinds of inductions from Spin(p,q) to Spin(p+1,q+1) are studied.

This article considers the coset structure of spin group via analyzing the expression of its representation. One example is also given to take a closer look at what the coset and the subgroup are.

We study the general representations of positive partial transpose (PPT) states in CK⊗CM⊗CN. For the PPT states with rank-N a canonical form is obtained, from which a sufficient separability condition is presented.

We give a canonical form of PPT states in C2 ⊗ C2 ⊗ C2 ⊗ CN with rank=N. From this canonical form a sufficient separability condition for these states is presented.

We consider rank-two density matrices ρ supported on an M × N Hilbert space for arbitrary dimensions M and N. Explicit sufficient and necessary conditions for separability of ρ are presented.

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.

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.

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.

We present the symplectic algorithm in the Lagrangian formalism for the Hamiltonian systems by virtue of the noncommutative differential calculus with respect to the discrete time and the Euler-Lagrange cohomological concepts. We also show that the trapezoidal integrator is symplectic in certain sense.

By means of a noncommutative differential calculus on function space of discrete Abelian groups and that of the regular lattice with equal spacing as well as the discrete symplectic geometry and a kind of classical mechanical systems with separable Hamiltonian of the type H(p, q) = T(p) + V(q) on regular lattice, we introduce the discrete symplectic algorithm, i.e., the phase-space discrete counterpart of the symplectic algorithm including original symplectic schemes and the jet-symplectic schemes in terms of the discrete time jet bundle formalism, on the regular lattice. We show some numerical calculation examples and compare the results of different schemes.

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.

We prove that the following three properties cannot match each other on a lattice, that differentials of coordinate functions are algebraically dependent on their involutive conjugates, that the involution on a lattice is an antihomomorphism, and that differential calculus has a natural continuum limit.

Using super RS construction method and Gauss decomposition, we obtain Drinfeld's currents realization of two-parameter quantum affine superalgebra $U_{p,q}\widehat{(gl(1|1))}$ and get co-product structure for these currents.

The N-fold Kerr family and charged Kerr family solutions have been given in this paper by means of the inductive method to solve the principal homogeneous Hilbert problems associated with the principal inverse scattering equations, i.e., the W-equations, for both vacuum and electrovac cases with stationary axial symmetry in general relativity.

It is shown that the integral of the k-th generalized secondary characteristic class, , over a manifold on which it is defined could be expressed as the one of an n-rank characteristic polynomial in the curvature of the interpolation connection over where is a k- simplex. And consequently the polynomial , say k=2, would be related to the analytic index of a certain elliptic operator on by means of the Atiyah-Patodi-Singer index theorem on and the adiabatic approximation.

We explain further the group property and algebra structure for a kind of Backlund transformations. If the Lax representation of a given nonlinear (evolution) system is dependent on a complex spectral parameter fractional linearly, the Backlund transformations with respect to the principal pseudopotentials (fundamental solutions) of the Lax representation can be identified with solutions of the regular matrix Riemann-Hilbert problem. It is shown that this kind of Riemann-Hilbert problem constructs an infinite dimensional Lie group and its infinitesimal form presents itself the infinite dimensional algebra of Kac-Moody type.

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.

An SL(2R) × R1(l) prolongation structure of Ernst equation with a real parameter l and the corresponding Riccati equation as well as a pair of linear equations which are in principle equivalent to the inverse scattering problem due to Belinsky and Zakharov are obtained by solving the fundamental equation for the prolongation structure. A necessary condition which should be satisfied by the Backlund transformations is presented in terms of prolongation structure. And it is indicated that in the, case of Ernst equation the Harrison transformation, Neuge-bauer transformations and other available Backlund transformations as well as Belinsky-Zakharov's Riemann transformation, i.e., the homogeneous Hilbert problem (HHP), would be covered by this condition.

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.

In addition to the recent discovery that the Chern-Simons secondary classes are the topological origins generating special unitary gauge group anomalies, it is shown that some special orthogonal gauge group anomalies can also be generated, from the Pontrjagin and the Euler secondary classes. The common differential geometrical and topological background are explained as well.

In the lecture notes[1] one of us (LU) has studied the relations between the theory of connections and Gauge theory. But the formulation of the connection theory or Gauge theory in the lecture notes is linear in nature, i.e., we mainly studied the covariant derivative of a vector, tensor or spinor fields. More precisely, given a differentiable fibre bundle with base space , fibre , structure group G and transition functions , where and belong to an open covering we say a connection defined on is linear if is a linear space and acts linearly on otherwise we say is non-linear.

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 faddeev's song's and zumino's approaches.

We will briefly review the prolongation structure theory for nonlinear systems, such as nonlinear evolution equations and soliton equations, and its applications to the solution generating techniques for the gravitational fields with two commuting Killing vectors.