Construction of Covariant Differential Calculi on Quantum Homogeneous Spaces

A method of constructing covariant differential calculi on a quantum homogeneous space is devised. The function algebra of the quantum homogeneous space is assumed to be a left coideal of a coquasitriangular Hopf algebra and to contain the coefficients of any matrix over which is the two-sided inverse of one with entries in . The method is based on partial derivatives. For the quantum sphere of Podle and the quantizations of symmetric spaces due to Noumi, Dijkhuizen and Sugitani, the construction produces the subcalculi of the standard bicovariant calculus on the quantum group.     

Hopf Modules and Noncommutative Differential Geometry

We define a new algebra of noncommutative differential forms for any Hopf algebra with an invertible antipode. We prove that there is a one-to-one correspondence between anti-Yetter–Drinfeld modules, which serve as coefficients for the Hopf cyclic (co)homology, and modules which admit a flat connection with respect to our differential calculus. Thus, we show that these coefficient modules can be regarded as “flat bundles” in the sense of Connes’ noncommutative differential geometry.     

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 multiparameter 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 erived 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.     

Differentials of Higher Order in Noncommutative Differential Geometry

In differential geometry, the notation dⁿ f along with the corresponding formalism has fallen into disuse since the birth of exterior calculus. However, differentials of higher-order are useful objects that can be interpreted in terms of functions on iterated tangent bundles (or in terms of jets). We generalize this notion to the case of noncommutative differential geometry. For an arbitrary algebra A, people already know how to define the differential algebra A of universal differential forms over A. We define Leibniz forms of order n (these are not forms of degree n, i.e. they are not elements of ⁿA) as particular elements of what we call the iterated frame algebra of order n, F_nA, which is itself defined as the2ⁿ tensor power of the algebra A. We give a system of generators for this iterated frame algebra and identify the A left-module of forms of order n as a particular vector subspace included in the space of universal 1-forms built over the iterated frame algebra of order n-1. We study the algebraic structure of these objects, recover the case of the commutative differential calculus of order n (Leibniz differentials) and give a few examples.     

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 Hamiltonican formalisms to get difference discrete Euler-Lagrange equations and canonical ones for the difference discrete versions of the classical mechanics and classical field theory.     

A Note on Symplectic Algorithm

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.``     

Cyclic Cohomology and Hopf Algebra Symmetry

Cyclic cohomology has been recently adapted to the treatment of Hopf symmetry in noncommutative geometry. The resulting theory of characteristic classes for Hopf algebras and their actions on algebras allows the expansion of the range of applications of cyclic cohomology. It is the goal of this Letter to illustrate these recent developments, with special emphasis on the application to transverse index theory, and point towards future directions. In particular, we highlight the remarkable accord between our framework for cyclic cohomology of Hopf algebras on the one hand and both the algebraic as well as the analytic theory of quantum groups on the other, manifest in the construction of the modular square.     

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

In the previous papers I and H, we have studied the difference discrete variational principle and the EulerLagrange cohomology in the framework of multi-parameter differential approach. W5 have gotten the difference discreteEulcr-Lagrangc equations and canonical ones for the difference discrete versions of classical mechanics and tield theoryas well as the difference discrete versions for the Euler-Lagrange cohomology and applied them to get the necessaryand sufficient condition for the symplectic or multisymplectic geometry preserving properties in both the Lagrangianand Hamiltonian formalisms. In this paper, we apply the difference discrete variational principle and Euler-Lagrangecohomological approach directly to the symplectic and multisymplectic algorithms. We will show that either Hamiltonianschemes or Lagrangian ones in both the symplectic and multisymplectic algorithms arc variational integrators and theirdifference discrete symplectic structure-preserving properties can always be established not only in the solution spacebut also in the function space if and only if the related closed Euler Lagrange cohomological conditions are satisfied.     

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 of 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.     

Deformations of Algebras Constructed using Quantum Stochastic Calculus

Deformations of associative algebras in which time is the deformation parameter are constructed using quantum stochastic flows in which the usual multiplicativity requirement is replaced by multiplicativity with respect to the deformed multiplication. The theory is restricted by a commutativity requirement on the structure maps of the flow, but examples which can be constructed in this way include the noncommutative torus and the Weyl–Moyal deformation.     

Noncommutative Cohomology and Electromagnetism on $$\mathbb{C}$$ q [SL2] at Roots of Unity

We compute the noncommutative de Rham cohomology for the finite dimensional q-deformed coordinate ring _q [SL₂] at odd roots of unity and with its standard four-dimensional differential structure. We find that H ⁱ is highly nontrivial compared to case of generic q and has as many modes 1 : 4 : 6 : 4 : 1 as the dimension of the exterior algebra. We solve the spin-0 and Maxwell theory on _q [SL₂], including a complete picture of the self-dual and anti-self dual solutions and of Lorentz and temporal gauge fixing. The system behaves in fact like a noncompact space with self-propagating modes (i.e., in the absence of sources). We also solve with examples of electric and magnetic sources including the biinvariant element H ¹ which we find can be viewed as a source in the local (Minkowski) time-direction (i.e. a uniform electric charge density).     

Noncommutative cumulants for stochastic differential equations and for generalized Dyson series

For the stochastic equationU=VU, Kubo's ansalze for U in the form of differential and integrodifferential equations is investigated and a newansatz as an integral equation is added. Unique solutions in terms of noncommutative W- and K-cumulants are found by elementary functional differentiation, and expressions of van Kämpen and Terwiel are recovered. For the cumulants we find simple recursion relations and prove the important cluster property. Surprisingly, it is found that the Gaussian approximation in the differential equationansatz leads to positivity problems, while this is not the case with the integral and integrodifferential equation. The cumulant expansion technique is carried over to generalized Dyson series. In a companion paper we apply our results to quantum shot noise.