On the definition of higher order differential operators in noncommutative geometry

Derivation based differential calculi for noncommutative algebras deforming a class of three dimensional spaces

We equip a family of algebras whose noncommutativity is of Lie type with a derivation based differential calculus obtained, upon suitably using both inner and outer derivations, as a reduction of a redundant calculus over the Moyal four dimensional space.

Noncommutative Euclidean spaces

We give a definition of noncommutative finite-dimensional Euclidean spaces $\mathbb R^n$. We then remind our definition of noncommutative products of Euclidean spaces $\mathbb R^{N_1}$ and $\mathbb R^{N_2}$ which produces noncommutative Euclidean spaces $\mathbb R^{N_1+N_2}$. We solve completely the conditions defining the noncommutative products of the Euclidean spaces $\mathbb R^{N_1}$ and $\mathbb R^{N_2}$ and prove that the corresponding noncommutative unit spheres $S^{N_1+N_2-1}$ are noncommutative spherical manifolds. We then apply these concepts to define "noncommutative" quaternionic p…

Vector Fields and Differential Operators: Noncommutative Case

A notion of Cartan pairs as an analogy of vector fields in the realm of noncommutative geometry has been proposed in q-alg/9609011 In this paper we give an outline of the construction of a noncommutative analogy of the algebra of partial differential operators as well as its natural (Fock type) representation. We shall also define co-universal vector fields and covariant derivatives.

On the notion of a differential operator in noncommutative geometry

The algebraic notion of a differential operator on a module over a commutative ring is not extended to a module over a noncommutative ring.

Comments on the Noncommutative Geometry of Quantum Homogeneous Vector Bundles

Differential calculi are obtained for quantum homogeneous spaces by extending Woronowicz' approach to the present context. Representation theoretical properties of the differential calculi are investigated. Connections on quantum homogeneous vector bundles are classified and explicitly constructed by using the theory of projective modules.

Noncommutative Martin-Lof randomness : on the concept of a random sequence of qubits

Martin-Lof's definition of random sequences of cbits as those not belonging to any set of constructive zero Lebesgue measure is reformulated in the language of Algebraic Probability Theory. The adoption of the Pour-El Richards theory of computability structures on Banach spaces allows us to give a natural noncommutative extension of Martin-Lof's definition, characterizing the random elements of a chain Von Neumann algebra. In the particular case of the minimally informative noncommutative alphabet our definition reduces to the definition of a random sequence of qubits.

Examples of derivation-based differential calculi related to noncommutative gauge theories

Some derivation-based differential calculi which have been used to construct models of noncommutative gauge theories are presented and commented. Some comparisons between them are made.

通过非交换Picard-Vessiot理论研究通用微分方程的解

On the solutions of universal differential equation by noncommutative Picard-Vessiot theory

本文通过非交换Picard-Vessiot理论提出递归构造通用微分方程解的方法，并应用分解法求解Knizhnik-Zamolodchikov方程的唯一解。

Levi-Civita connections and vector fields for noncommutative differential calculi

We study covariant derivatives on a class of centered bimodules $\mathcal{E}$ over an algebra A. We begin by identifying a $\mathbb{Z} ( A ) $-submodule $ \mathcal{X} ( A ) $ which can be viewed as the analogue of vector fields in this context; $ \mathcal{X} ( A ) $ is proven to be a Lie algebra. Connections on $\mathcal{E}$ are in one to one correspondence with covariant derivatives on $ \mathcal{X} ( A ). $ We recover the classical formulas of torsion and metric compatibility of a connection in the covariant derivative form. As a result, a Koszul formula for the Levi-Civita connection is als…

Jets and connections in commutative and noncommutative geometry

It is emphasized that equivalent definitions of connections on modules over commutative rings are not so in noncommutative geometry.

Kronecker代数的非交换黎曼几何

Noncommutative Riemannian geometry of Kronecker algebras

本文研究Kronecker箭图路径代数的非交换黎曼几何，构造无挠双模联络及Levi-Civita联络，并探讨其与厄米形式的相容性。

Realization of bicovariant differential calculus on the Lie algebra type noncommutative spaces

This paper investigates bicovariant differential calculus on noncommutative spaces of the Lie algebra type. For a given Lie algebra $g_0$ we construct a Lie superalgebra $g=g_0\oplus g_1$ containing noncommutative coordinates and one--forms. We show that $g$ can be extended by a set of generators $T_{AB}$ whose action on the enveloping algebra $U(g)$ gives the commutation relations between monomials in $U(g_0)$ and one--forms. Realizations of noncommutative coordinates, one--forms and the generators $T_{AB}$ as formal power series in a semicompleted Weyl superalgebra are found. In the special …

Jets of modules in noncommutative geometry

Jets of modules over a commutative ring are well known to make up the representative objects of linear differential operators on these modules. In noncommutative geometry, jets of modules provide the representative objects only of a certain class of first order differential operators. As a consequence, a generalization of the standard Lagrangian formalism on smooth manifolds to noncommutative spaces is problematic.

An Example of $Z_{N}$-Graded Noncommutative Differential Calculus

In this work, we consider the algebra $M_{N}(C)$ of $N\times N$ matrices as a cyclic quantum plane. We also analyze the coaction of the quantum group ${\cal F}$ and the action of its dual quantum algebra ${\cal H}$ on it. Then, we study the decomposition of $M_{N}(C)$ in terms of the quantum algebra representations. Finally, we develop the differential algebra of the cyclic group $Z_{N}$ with $d^{N}=0$, and treat the particular case N=3.

Differential Forms and the Noncommutative Residue for Manifolds with Boundary in the Non-product Case

In this paper, for an even dimensional compact manifold with boundary which has the non-product metric near the boundary, we use the noncommutative residue to define a conformal invariant pair. For a 4-dimensional manifold, we compute this conformal invariant pair under some conditions and point out the way of computations in the general.

Curvature and geometric modules of noncommutative spheres and tori

When considered as submanifolds of Euclidean space, the Riemannian geometry of the round sphere and the Clifford torus may be formulated in terms of Poisson algebraic expressions involving the embedding coordinates, and a central object is the projection operator, projecting tangent vectors in the ambient space onto the tangent space of the submanifold. In this note, we point out that there exist noncommutative analogues of these projection operators, which implies a very natural definition of noncommutative tangent spaces as particular projective modules. These modules carry an induced connec…

Noncommutative Differential Geometry and Classical Field Theory on Finite Groups

Plan of this report is given below: 1. Motivation from Physical and Mathematical Point of View; 2. Differential Calculi on Finite Groups; 3. Metrics; 4. Lagrangian Field Theory and Symplectic Structure; 5. Scalar Field Theory and Spectral of Finite Groups.

Noncommutative circle bundles and new Dirac operators

We study spectral triples over noncommutative principal U(1) bundles. Basing on the classical situation and the abstract algebraic approach, we propose an operatorial definition for a connection and compatibility between the connection and the Dirac operator on the total space and on the base space of the bundle. We analyze in details the example of the noncommutative three-torus viewed as a U(1) bundle over the noncommutative two-torus and find all connections compatible with an admissible Dirac operator. Conversely, we find a family of new Dirac operators on the noncommutative tori, which ar…

Noncommutative Geometry of Phase Space

We investigate the geometric, algebraic and homologic structures related with Poisson structure on a smooth manifold. Introduce a noncommutative foundations of these structures for a Poisson algebra. Introduce and investigate noncommutative Bott connection on a foliated manifold using the algebraic definition of submanifold and quotient manifold. Develop an algebraic construction for the reduction of a degenerated Poisson algebra.

非交换微分形式上Levi-Civita联络的存在性与唯一性

Existence and uniqueness of the Levi-Civita connection on noncommutative differential forms

该研究结合希尔伯特模与代数方法，给出非交换微分形式上埃尔米特无挠联络的存在充要条件与唯一充分条件，证明紧黎曼流形θ-形变存在唯一该类联络并显式构造，还得到经典Levi-Civita联络的新构造。

Dirac-Kaehler Fermion from Clifford Product with Noncommutative Differential Form on a Lattice

We formulate Dirac-Kaehler fermion action by introducing a new Clifford product with noncommutative differential form on a lattice. Hermiticity of the Dirac-Kaehler action requires to choose the lattice structure having both orientabilities on a link. The Kogut-Susskind fermion and the staggered fermion actions are derived directly from the Dirac-Kaehler fermion formulated by the Clifford product. The lattice QCD action with Dirac-Kaehler matter fermion is also derived via an inner product defined by the Clifford product.

Noncommutative Yang-Mills-Higgs actions from derivation-based differential calculus

Derivations of a noncommutative algebra can be used to construct differential calculi, the so-called derivation-based differential calculi. We apply this framework to a version of the Moyal algebra ${\cal{M}}$. We show that the differential calculus, generated by the maximal subalgebra of the derivation algebra of ${\cal{M}}$ that can be related to infinitesimal symplectomorphisms, gives rise to a natural construction of Yang-Mills-Higgs models on ${\cal{M}}$ and a natural interpretation of the covariant coordinates as Higgs fields. We also compare in detail the main mathematical properties ch…

Lectures on Differential Geometry of Modules and Rings

Generalizing differential geometry of smooth vector bundles formulated in algebraic terms of the ring of smooth functions, its derivations and the Koszul connection, one can define differential operators, differential calculus and connections on modules over arbitrary commutative, graded commutative and noncommutative rings. For instance, this is the case of quantum theory, SUSY theory and noncommutative geometry, respectively. The relevant material on this subject is summarized.

量子演算：非交换几何中的微分形式与向量场

Quantum Calculi: differential forms and vector fields in noncommutative geometry

本文修订并扩展非交换向量场框架，填补旧有不足，阐明嘉当对与一阶微分的对应，并显式构造右/左泛向量场双模。

Harmonic analysis on homogeneous complex bounded domains and noncommutative geometry

We define and study a noncommutative Fourier transform on every homogeneous complex bounded domain. We then give an application in noncommutative differential geometry by defining noncommutative Baumslag-Solitar tori.

Noncommutative Ricci flow in a matrix geometry

We study noncommutative Ricci flow in a finite dimensional representation of a noncommutative torus. It is shown that the flow exists and converges to the flat metric. We also consider the evolution of entropy and a definition of scalar curvature in terms of the Ricci flow.

非对易几何中的比安基恒等式

Bianchi identities in noncommutative geometry

本文研究带有三角Hopf代数表示的非对易代数的微分几何，提出任意联络比安基恒等式的新方法，证明其等价性并得到广义相对论比安基恒等式的非对易类似物。

Noncommutative epsilon-graded connections

We introduce the new notion of epsilon-graded associative algebras which takes its root into the notion of commutation factors introduced in the context of Lie algebras. We define and study the associated notion of epsilon-derivation-based differential calculus, which generalizes the derivation-based differential calculus on associative algebras. A corresponding notion of noncommutative connection is also defined. We illustrate these considerations with various examples of epsilon-graded algebras, in particular some graded matrix algebras and the Moyal algebra. This last example permits also t…

无穷小空间上的非交换微分几何

Noncommutative Differential Geometry on Infinitesimal Spaces

本文利用非交换微分几何，通过偏序集上的 $C^*$-代数与有限维谱三元组形式化离散微分演算，并证明格点与环面情形下的收敛性。

Noncommutative Geometry of Lattice and Staggered Fermions

Differential structure of a d-dimensional lattice, which is essentially a noncommutative exterior algebra, is defined using reductions in first order and second order of universal differential calculus in the context of noncommutative geometry (NCG) developed by Dimakis et al. This differential structure can be realized adopting a Dirac-Connes operator proposed by us recently within Connes' NCG. With matrix representations being specified, our Dirac-Connes operator corresponds to staggered Dirac operator, in the case that dimension of the lattice equals to 1, 2 and 4.

Noncommutative coordinate algebras

We discuss the noncommutative generalizations of polynomial algebras which after appropriate completions can be used as coordinate algebras in various noncommutative settings, (noncommutative differential geometry, noncommutative algebraic geometry, etc.). These algebras have finite presentations and are completely characterized and classified by their (noncommutative) volume forms.

The noncommutative geometry of the Landau Hamiltonian: Differential aspects

In this work we study the differential aspects of the noncommutative geometry for the magnetic $C^*$-algebra which is a 2-cocycle deformation of the group $C^*$-algebra of $\mathbb{R}^2$. This algebra is intimately related to the study of the Quantum Hall Effect in the continuous, and our results aim to provide a new geometric interpretation of the related Kubo's formula. Taking inspiration from the ideas developed by Bellissard during the 80's, we build an appropriate Fredholm module for the magnetic $C^*$-algebra based on the magnetic Dirac operator which is the square root (à la Dirac) of t…

Translation-Invariant Noncommutative Gauge Theories, Matrix Modeling and Noncommutative Geometry

A matrix modeling formulation for translation-invariant noncommutative gauge theories is given in the setting of differential graded algebras and quantum groups. Translation-invariant products are discussed in the setting of α-cohomology and it is shown that loop calculations are entirely determined by α-cohomology class of star product in all orders. Noncommutative version of geometric quantization and (anti-) BRST transformations is worked out which leads to a noncommutative description of consistent anomalies and Schwinger terms.

非交换离散方程、对称性与约化

Noncommutative discrete equations, symmetries and reductions

本文利用非交换离散KdV方程的Lax对构造广义对称与Miura变换，通过对称约化得到非交换离散Painlevé方程和推广的Ernst方程，并建立Darboux变换、自Bäcklund变换与非交换Yang–Baxter映射的联系。

A note on the implementation of Poincaré symmetry in noncommutative field theory

We argue that Poincaré symmetry can be implemented in NCFT if we allow the parameter of noncommutitive deformation $θ^{μν}$ to change as a two-tensor under the corresponding space-time symmetry. The implementation is consistent with the definition of $θ^{μν}$ in terms of space-time coordinates and with the Moyal star product. Inspired from the standard definition of a variational symmetry we found a universal way to correct the implementation of the Poincaré symmetry by a term proportional to the variation of $θ^{μν}$ in such a way that the new transformation define a symmetry of the theory. F…

Automorphisms of associative algebras and noncommutative geometry

A class of differential calculi is explored which is determined by a set of automorphisms of the underlying associative algebra. Several examples are presented. In particular, differential calculi on the quantum plane, the $h$-deformed plane and the quantum group GLpq(2) are recovered in this way. Geometric structures like metrics and compatible linear connections are introduced.