8 research outputs found
Discrete Riemannian Geometry
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.Comment: 34 pages, 1 figure (eps), LaTeX, amssymb, epsfi
Differential Calculi on Commutative Algebras
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.Comment: 33 pages, LaTeX. Revision: A remark about a quasilattice and Penrose
tiling was incorrect in the first version of the paper (p. 14
Explicit Flow Equations and Recursion Operator of the ncKP hierarchy
The explicit expression of the flow equations of the noncommutative
Kadomtsev-Petviashvili(ncKP) hierarchy is derived. Compared with the flow
equations of the KP hierarchy, our result shows that the additional terms in
the flow equations of the ncKP hierarchy indeed consist of commutators of
dynamical coordinates \{\}. The recursion operator for the flow equations
under -reduction is presented. Further, under 2-reduction, we calculate a
nonlocal recursion operator of the noncommutative Korteweg-de
Vries(ncKdV) hierarchy, which generates a hierarchy of local, higher-order
flows. Thus we solve the open problem proposed by P.J. Olver and V.V.
Sokolov(Commun.Math.Phys. 193 (1998), 245-268).Comment: 20pages,no figure, accepted by Nonlinearity(Aug., 2011
Factorization methods for Noncommutative KP and Toda hierarchy
We show that the solution space of the noncommutative KP hierarchy is the
same as that of the commutative KP hierarchy owing to the Birkhoff
decomposition of groups over the noncommutative algebra. The noncommutative
Toda hierarchy is introduced. We derive the bilinear identities for the
Baker--Akhiezer functions and calculate the -soliton solutions of the
noncommutative Toda hierarchy.Comment: 7 pages, no figures, AMS-LaTeX, minor corrections, final version to
appear in Journal of Physics
Noncommutative Burgers Equation
We present a noncommutative version of the Burgers equation which possesses
the Lax representation and discuss the integrability in detail. We find a
noncommutative version of the Cole-Hopf transformation and succeed in the
linearization of it. The linearized equation is the (noncommutative) diffusion
equation and exactly solved. We also discuss the properties of some exact
solutions. The result shows that the noncommutative Burgers equation is
completely integrable even though it contains infinite number of time
derivatives. Furthermore, we derive the noncommutative Burgers equation from
the noncommutative (anti-)self-dual Yang-Mills equation by reduction, which is
an evidence for the noncommutative Ward conjecture. Finally, we present a
noncommutative version of the Burgers hierarchy by both the Lax-pair generating
technique and the Sato's approach.Comment: 24 pages, LaTeX, 1 figure; v2: discussions on Ward conjecture, Sato
theory and the integrability added, references added, version to appear in J.
Phys.