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.
The Chevalley-Eilenberg differential calculus and differential operators over N-graded commutative rings are constructed. This is a straightforward generalization of the differential calculus over commutative rings, and it is the most general case of the differential calculus over rings that is not the non-commutative geometry. Since any N-graded ring possesses the associated Z_2-graded structure, this also is the case of the graded differential calculus over Grassmann algebras and the supergeometry and field theory on graded manifolds.
We develop a $GL_{qp}(2)$ invariant differential calculus on a two-dimensional noncommutative quantum space. Here the co-ordinate space for the exterior quantum plane is spanned by the differentials that are commutative (bosonic) in nature.
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 …
These notes contain a survey of some aspects of the theory of graded differential algebras and of noncommutative differential calculi as well as of some applications connected with physics. They also give a description of several new developments.
In this thesis, we studied certain mathematical issues related to the computation of the Chamseddine--Connes spectral action on some fundamental noncommutative spectral triples, such as the noncommutative torus and the quantum 3-sphere SUq(2). We showed in particular that a Diophantine condition on the deformation matrix of the torus is crucial to obtain the spectral action with real structure. We also studied the question of existence of tadpoles (linear terms in the gauge potential of the fluctuation of the metric in the spectral action) for commutative Riemannian geometries, and the constru…
We study the graded derivation-based noncommutative differential geometry of the $Z_2$-graded algebra ${\bf M}(n| m)$ of complex $(n+m)\times(n+m)$-matrices with the ``usual block matrix grading'' (for $n\neq m$). Beside the (infinite-dimensional) algebra of graded forms the graded Cartan calculus, graded symplectic structure, graded vector bundles, graded connections and curvature are introduced and investigated. In particular we prove the universality of the graded derivation-based first-order differential calculus and show, that ${\bf M}(n|m)$ is a ``noncommutative graded manifold'' in a st…
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…
In this paper we study the Cauchy problem for diffusion equations associated to a class of strongly hypoelliptic pseudo-differential operators on graded Lie groups. To do so, we develop a global complex functional calculus on graded Lie groups in order to analyse the corresponding energy estimates. One of the main aspects of this complex functional calculus is that for the $(ρ,δ)$-Euclidean Hörmander classes we recover the standard functional calculus developed by Seeley [38]. In consequence the Gårding inequality that we prove for arbitrary graded Lie groups absorbs the historical 1953's ineq…
We present a possible generalization of the exterior differential calculus, based on the operator d such that d^3=0, but d^2\not=0. The first and second order differentials generate an associative algebra; we shall suppose that there are no binary relations between first order differentials, while the ternary products will satisfy the cyclic relations based on the representation of cyclic group Z_3 by cubic roots of unity. We shall attribute grade 1 to the first order differentials and grade 2 to the second order differentials; under the associative multiplication law the grades add up modulo …
We study the continuity property of multiple Q-adapted quantum stochastic integrals with respect to noncommuting integrands given by the non-adapted multiple integral kernels in Fock scale. The noncommutative algebra of relatively (exponentially) bounded nonadapted quantum stochastic processes is studied in the kernel form as introduced by Belavkin in 1991. The differential Q-adapted formula generalizing Ito product formula for adapted integrals is presented in both strong and weak sense as a particular case of the quantum stochastic nonadapted Ito formula.
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…
Richard Borcherds proposed an elegant geometric version of renormalized perturbative quantum field theory in curved spacetimes, where Lagrangians are sections of a Hopf algebra bundle over a smooth manifold. However, this framework looses its geometric meaning when Borcherds introduces a (graded) commutative normal product. We present a fully geometric version of Borcherds' quantization where the (external) tensor product plays the role of the normal product. We construct a noncommutative many-body Hopf algebra and a module over it which contains all the terms of the perturbative expansion and…
In noncommutative spaces, it is unknown whether the Pontrjagin class gives integer, as well as, the relation between the instanton number and Pontrjagin class is not clear. Here we define ``Instanton number'' by the size of $B_α$ in the ADHM construction. We show the analytical derivation of the noncommuatative U(1) instanton number as an integral of Pontrjagin class (instanton charge) with the Fock space representation. Our approach is for the arbitrary converge noncommutative U(1) instanton solution, and is based on the anti-self-dual (ASD) equation itself. We give the Stokes' theorem for th…
In order to investigate to what extent the calculus of classical (pseudo-)Riemannian manifolds can be extended to a noncommutative setting, we introduce pseudo-Riemannian calculi of modules over noncommutative algebras. In this framework, it is possible to prove an analogue of Levi-Civita's theorem, stating that there exists at most one torsion-free and metric connection for a given (metric) module, satisfying the requirements of a real metric calculus. Furthermore, the corresponding curvature operator has the same symmetry properties as the classical Riemannian curvature. As our main motivati…
The relevant material on differential calculus on graded infinite order jet manifolds and its cohomology is summarized. This mathematics provides the adequate formulation of Lagrangian theories of even and odd variables on smooth manifolds in terms of the Grassmann-graded variational bicomplex.
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.
The Leibniz rule for derivations is invariant under cyclic permutations of co-multiples within the arguments of derivations. We explore the implications of this principle: in effect, we construct a class of noncommutative bundles in which the sheaves of algebras of walks along a tesselated affine manifold form the base, whereas the fibres are free associative algebras or, at a later stage, such algebras quotients over the linear relation of equivalence under cyclic shifts. The calculus of variations is developed on the infinite jet spaces over such noncommutative bundles. In the frames of such…
Derivation-based differential calculi are of great importance in noncommutative geometry, noncommutative gauge theory and integrable systems. In this paper, we propose the connection and curvature from a class of deformed derivation-based differential calculus. By means of this theory, we give out the zero-curvature representation of the continuum-continuum, discrete-continuum and discrete-discrete integrable systems in an unifying manner.
The differential calculus based on the derivations of an associative algebra underlies most of the noncommutative field theories considered so far. We review the essential properties of this framework and the main features of noncommutative connections in the case of non graded associative unital algebras with involution. We extend this framework to the case of ${\mathbb{Z}}_2$-graded unital involutive algebras. We show, in the case of the Moyal algebra or some related ${\mathbb{Z}}_2$-graded version of it, that the derivation based differential calculus is a suitable framework to construct Ya…
It is well known that for a given Poisson structure one has infinitely many star products related through the Kontsevich gauge transformations. These gauge transformations have an infinite functional dimension (i.e., correspond to an infinite number of degrees of freedom per point of the base manifold). We show that on a symplectic manifold this freedom may be almost completely eliminated if one extends the star product to all tensor fields in a covariant way and impose some natural conditions on the tensor algebra. The remaining ambiguity either correspond to constant renormalizations to the …
We prove $L_{\infty}$-formality for the higher cyclic Hochschild complex $\chH$ over free associative algebra or path algebra of a quiver. The $\chH$ complex is introduced as an appropriate tool for the definition of pre-Calabi-Yau structure. We show that cohomologies of this complex are pure in case of free algebras (path algebras), concentrated in degree zero. It serves as a main ingredient for the formality proof. For any smooth algebra we choose a small qiso subcomplex in the higher cyclic Hochschild complex, which gives rise to a calculus of highly noncommutative monomials, we call them $…
We combine the coordinate method and Erlangen program in the framework of noncommutative geometry through an investigation of symmetries of noncommutative coordinate algebras. As the model we use the coherent states construction and the wavelet transform in functional spaces. New examples are a three dimensional spectrum of a non-normal matrix and a quantisation procedure from symplectomorphisms. Keywords: Heisenberg group, special linear group, symplectic group, Hardy space, Segal-Bargmann space, Clifford algebra, Cauchy-Riemann-Dirac operator, Möbius transformations, functional calculus, Wey…
In contrast with differential operators on modules over commutative and graded commutative rings, there is no satisfactory notion of a differential operator in noncommutative geometry.
In this paper we introduce a kind of "noncommutative neighbourhood" of a semiclassical parameter corresponding to the Planck constant. This construction is defined as a certain filtered and graded algebra with an infinite number of generators indexed by planar binary leaf-labelled trees. The associated graded algebra (the classical shadow) is interpreted as a "distortion" of the algebra of classical observables of a physical system. It is proven that there exists a q-analogue of the Weyl quantization, where q is a matrix of formal variables, which induces a nontrivial noncommutative analogue o…
We construct a noncommutative Cartan calculus on any braided commutative algebra and study its applications in noncommutative geometry. The braided Lie derivative, insertion and de Rham differential are introduced and related via graded braided commutators, also incorporating the braided Schouten-Nijenhuis bracket. The resulting braided Cartan calculus generalizes the Cartan calculus on smooth manifolds and the twisted Cartan calculus. While it is a necessity of derivation based Cartan calculi on noncommutative algebras to employ central bimodules our approach allows to consider bimodules over…
We generalize the notion, introduced by Henri Cartan, of an operation of a Lie algebra $\mathfrak g$ in a graded differential algebra $Ω$. We define the notion of an operation of a Hopf algebra $\mathcal H$ in a graded differential algebra $Ω$ which is refered to as a $\mathcal H$-operation. We then generalize for such an operation the notion of algebraic connection. Finally we discuss the corresponding noncommutative version of the Weil algebra: The Weil algebra $W(\mathcal H)$ of the Hopf algebra $\mathcal H$ is the universal initial object of the category of $\mathcal H$-operations with con…
We construct a differential calculus over the noncommutative 4-sphere in the framework of pseudo-Riemannian calculi, and show that for every metric in a conformal class of perturbations of the round metric, there exists a unique metric and torsion-free connection. Furthermore, we find a localization of the projective module corresponding to the space of vector fields, which allows us to formulate a Gauss-Chern-Bonnet type theorem for the noncommutative 4-sphere.
We study quantum Dirichlet forms and the associated symmetric quantum Markov semigroups on noncommutative $L^2$ spaces. It is known from the work of Cipriani and Sauvageot that these semigroups induce a first order differential calculus, and we use this differential calculus to define a noncommutative transport metric on the set of density matrices. This construction generalizes both the $L^2$-Wasserstein distance on a large class of metric spaces as well as the discrete transport distance introduced by Maas, Mielke, and Chow-Huang-Li-Zhou. Assuming a Bakry-Émery-type gradient estimate, we sho…
