arXiv subjects
Publications and source records attributed to A. Dimakis.
The automorphisms of all 4-dimensional, real Lie Algebras are presented in a comprehensive way. Their action on the space of $4\times 4$, real, symmetric and positive definite, matrices, defines equivalence classes which are used for the invariant characterization of the 4-dimensional homogeneous spaces which possess an invariant basis.
By considering a lattice model of extended phase space, and using techniques of noncommutative differential geometry, we are led to: (a) the conception of vector fields as generators of motion and transition probability distributions on the lattice; (b) the emergence of the time direction on the basis of the encoding of probabilities in the lattice structure; (c) the general prescription for the observables' evolution in analogy with classical dynamics. We show that, in the limit of a continuous description, these results lead to the time evolution of observables in terms of (the adjoint of) generalized Fokker-Planck equations having: (1) a diffusion coefficient given by the limit of the correlation matrix of the lattice coordinates with respect to the probability distribution associated with the generator of motion; (2) a drift term given by the microscopic average of the dynamical equations in the present context. These results are applied to 1D and 2D problems. Specifically, we derive: (I) The equations of diffusion, Smoluchowski and Fokker-Planck in velocity space, thus indicating the way random walk models are incorporated in the present context; (II) Kramers' equation, by further assuming that, motion is deterministic in coordinate space
We introduce a Hodge operator in a framework of noncommutative geometry. The complete integrability of 2-dimensional classical harmonic maps into groups (sigma-models or principal chiral models) is then extended to a class of 'noncommutative' harmonic maps into matrix algebras.
Within a framework of noncommutative geometry, we develop an analogue of (pseudo) Riemannian geometry on finite and discrete sets. On a finite set, there is a counterpart of the continuum metric tensor with a simple geometric interpretation. The latter is based on a correspondence between first order differential calculi and digraphs. Arrows originating from a vertex span its (co)tangent space. If the metric is to measure length and angles at some point, it has to be taken as an element of the left-linear tensor product of the space of 1-forms with itself, and not as an element of the (non-local) tensor product over the algebra of functions. It turns out that linear connections can always be extended to this left tensor product, so that metric compatibility can be defined in the same way as in continuum Riemannian geometry. In particular, in the case of the universal differential calculus on a finite set, the Euclidean geometry of polyhedra is recovered from conditions of metric compatibility and vanishing torsion. In our rather general framework (which also comprises structures which are far away from continuum differential geometry), there is in general nothing like a Ricci tensor or a curvature scalar. Because of the non-locality of tensor products (over the algebra of functions) of forms, corresponding components (with respect to some module basis) turn out to be rather non-local objects. But one can make use of the parallel transport associated with a connection to `localize' such objects and in certain cases there is a distinguished way to achieve this. This leads to covariant components of the curvature tensor which then allow a contraction to a Ricci tensor. In the case of a differential calculus associated with a hypercubic lattice we propose a new discrete analogue of the (vacuum) Einstein equations.
A generalization of the notion of a (pseudo-) Riemannian space is proposed in a framework of noncommutative geometry. In particular, there are parametrized families of generalized Riemannian spaces which are deformations of classical geometries. We also introduce harmonic maps on generalized Riemannian spaces into Hopf algebras and make contact with integrable models in two dimensions.
An introduction is given to some selected aspects of noncommutative geometry. Simple examples in this context are provided by finite sets and lattices. As an application, it is explained how the nonlinear Toda lattice and a discrete time version of it can be understood as generalized sigma-models based on noncommutative geometries. In particular, in this way one achieves a simple understanding of the complete integrability of the Toda lattice. Furthermore, generalized metric structures on finite sets and lattices are briefly discussed.
Familiar nonlinear and in particular soliton equations arise as zero curvature conditions for GL(1,R) connections with noncommutative differential calculi. The Burgers equation is formulated in this way and the Cole-Hopf transformation for it attains the interpretation of a transformation of the connection to a pure gauge in this mathematical framework. The KdV, modified KdV equation and the Miura transformation are obtained jointly in a similar setting and a rather straightforward generalization leads to the KP and a modified KP equation. Furthermore, a differential calculus associated with the Boussinesq equation is derived from the KP calculus.
In the context of a noncommutative differential calculus on the algebra of real valued functions of an $n$-dimensional manifold $M$, a commutative and associative product of 1-forms is naturally defined. Ordinary differential calculus corresponds to this product being trivially zero. We consider the minimal generalization in which the algebra of 1-forms with this product is nilpotent of degree 3. Basic tensor analysis and differential geometry are developped in this context and applied to the formulation of symplectic geometry and Hamiltonian dynamics. It is shown that the corresponding Liouville equation can take the form of a generalized Fokker-Planck equation, well known in statistical mechanics of open systems, as well as in stochastic calculus. Specifically it may be the same with that obtained via Ito stochastic calculus, when solving a linear stochastic differential equation with a Wiener process as the stochastic term. The close connection between noncommutative differential structures of this kind, with semimartingale stochastic processes on manifolds thus suggested, is further explored.
It is shown that the only associative algebras with a trivial center defined on functions of $\Rl^N$ by an integral kernel are generalized Moyal algebras, corresponding to some particular operator ordering. Similarly, the only such Lie algebras are generalized Moyal or Poisson Lie algebras. In both cases these structures are isomorphic respectively to the Moyal algebra, the Moyal Lie algebra and the Poisson Lie algebra. These results are independent of $N$, which is necessarily even, and generalize previous work on the subject.
A construction of conservation laws for $σ$-models in two dimensions is generalized in the framework of noncommutative geometry of commutative algebras. This is done by replacing the ordinary calculus of differential forms with other differential calculi and introducing an analogue of the Hodge operator on the latter. The general method is illustrated with several examples.
A construction of conservation laws for chiral models (generalized sigma-models on a two-dimensional space-time continuum using differential forms is extended in such a way that it also comprises corresponding discrete versions. This is achieved via a deformation of the ordinary differential calculus. In particular, the nonlinear Toda lattice results in this way from the linear (continuum) wave equation. The method is applied to several further examples. We also construct Lax pairs and Bäcklund transformations for the class of models considered in this work.
`Umbral calculus' deals with representations of the canonical commutation relations. We present a short exposition of it and discuss how this calculus can be used to discretize continuum models and to construct representations of Lie algebras on a lattice. Related ideas appeared in recent publications and we show that the examples treated there are special cases of umbral calculus. This observation then suggests various generalizations of these examples. A special umbral representation of the canonical commutation relations given in terms of the position and momentum operator on a lattice is investigated in detail.
A finite set can be supplied with a group structure which can then be used to select (classes of) differential calculi on it via the notions of left-, right- and bicovariance. A corresponding framework has been developed by Woronowicz, more generally for Hopf algebras including quantum groups. A differential calculus is regarded as the most basic structure needed for the introduction of further geometric notions like linear connections and, moreover, for the formulation of field theories and dynamics on finite sets. Associated with each bicovariant first order differential calculus on a finite group is a braid operator which plays an important role for the construction of distinguished geometric structures. For a covariant calculus, there are notions of invariance for linear connections and tensors. All these concepts are explored for finite groups and illustrated with examples. Some results are formulated more generally for arbitrary associative (Hopf) algebras. In particular, the problem of extension of a connection on a bimodule (over an associative algebra) to tensor products is investigated, leading to the class of `extensible connections'. It is shown that invariance properties of an extensible connection on a bimodule over a Hopf algebra are carried over to the extension. Furthermore, an invariance property of a connection is also shared by a `dual connection' which exists on the dual bimodule (as defined in this work).
The basic mathematical assumptions for autonomous linear kinetic equations for a classical system are formulated, leading to the conclusion that if they are differential equations on its phase space $M$, they are at most of the 2nd order. For open systems interacting with a bath at canonical equilibrium they have a particular form of an equation of a generalized Fokker-Planck type. We show that it is possible to obtain them as Liouville equations of Hamiltonian dynamics on $M$ with a particular non-commutative differential structure, provided certain geometric in character, conditions are fulfilled. To this end, symplectic geometry on $M$ is developped in this context, and an outline of the required tensor analysis and differential geometry is given. Certain questions for the possible mathematical interpretation of this structure are also discussed.
A differential calculus on an associative algebra A is an algebraic analogue of the calculus of differential forms on a smooth manifold. It supplies A with a structure on which dynamics and field theory can be formulated to some extent in very much the same way we are used to from the geometrical arena underlying classical physical theories and models. In previous work, certain differential calculi on a commutative algebra exhibited relations with lattice structures, stochastics, and parametrized quantum theories. This motivated the present systematic investigation of differential calculi on commutative and associative algebras. Various results about their structure are obtained. In particular, it is shown that there is a correspondence between first order differential calculi on such an algebra and commutative and associative products in the space of 1-forms. An example of such a product is provided by the Ito calculus of stochastic differentials. For the case where the algebra A is freely generated by `coordinates' x^i, i=1,...,n, we study calculi for which the differentials dx^i constitute a basis of the space of 1-forms (as a left A-module). These may be regarded as `deformations' of the ordinary differential calculus on R^n. For n < 4 a classification of all (orbits under the general linear group of) such calculi with `constant structure functions' is presented. We analyse whether these calculi are reducible (i.e., a skew tensor product of lower-dimensional calculi) or whether they are the extension (as defined in this article) of a one dimension lower calculus. Furthermore, generalizations to arbitrary n are obtained for all these calculi.
Differential calculus on discrete sets is developed in the spirit of noncommutative geometry. Any differential algebra on a discrete set can be regarded as a `reduction' of the `universal differential algebra' and this allows a systematic exploration of differential algebras on a given set. Associated with a differential algebra is a (di)graph where two vertices are connected by at most two (antiparallel) arrows. The interpretation of such a graph as a `Hasse diagram' determining a (locally finite) topology then establishes contact with recent work by other authors in which discretizations of topological spaces and corresponding field theories were considered which retain their global topological structure. It is shown that field theories, and in particular gauge theories, can be formulated on a discrete set in close analogy with the continuum case. The framework presented generalizes ordinary lattice theory which is recovered from an oriented (hypercubic) lattice graph. It also includes, e.g., the two-point space used by Connes and Lott (and others) in models of elementary particle physics. The formalism suggests that the latter be regarded as an approximation of a manifold and thus opens a way to relate models with an `internal' discrete space ({à} la Connes et al.) to models of dimensionally reduced gauge fields. Furthermore, also a `symmetric lattice' is studied which (in a certain continuum limit) turns out to be related to a `noncommutative differential calculus' on manifolds.
A `discrete differential manifold' we call a countable set together with an algebraic differential calculus on it. This structure has already been explored in previous work and provides us with a convenient framework for the formulation of dynamical models on networks and physical theories with discrete space and time. We present several examples and introduce a notion of differentiability of maps between discrete differential manifolds. Particular attention is given to differentiable curves in such spaces. Every discrete differentiable manifold carries a topology and we show that differentiability of a map implies continuity.
There is a deformation of the ordinary differential calculus which leads from the continuum to a lattice (and induces a corresponding deformation of physical theories). We recall some of its features and relate it to a general framework of differential calculus on discrete sets. This framework generalizes the usual (lattice) discretization.