40 research outputs found
Discrete differential calculus, graphs, topologies and gauge theory
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 ({\`a} 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.Comment: 36 pages, revised version, appendix adde
Bi-differential calculus and the KdV equation
A gauged bi-differential calculus over an associative (and not necessarily
commutative) algebra A is an N-graded left A-module with two covariant
derivatives acting on it which, as a consequence of certain (e.g., nonlinear
differential) equations, are flat and anticommute. As a consequence, there is
an iterative construction of generalized conserved currents. We associate a
gauged bi-differential calculus with the Korteweg-de-Vries equation and use it
to compute conserved densities of this equation.Comment: 9 pages, LaTeX, uses amssymb.sty, XXXI Symposium on Mathematical
Physics, Torun, May 1999, replaces "A notion of complete integrability in
noncommutative geometry and the Korteweg-de-Vries equation
Multicomponent Burgers and KP Hierarchies, and Solutions from a Matrix Linear System
Via a Cole-Hopf transformation, the multicomponent linear heat hierarchy leads to a multicomponent Burgers hierarchy. We show in particular that any solution of the latter also solves a corresponding multicomponent (potential) KP hierarchy. A generalization of the Cole-Hopf transformation leads to a more general relation between the multicomponent linear heat hierarchy and the multicomponent KP hierarchy. From this results a construction of exact solutions of the latter via a matrix linear system
A new approach to deformation equations of noncommutative KP hierarchies
Partly inspired by Sato's theory of the Kadomtsev-Petviashvili (KP)
hierarchy, we start with a quite general hierarchy of linear ordinary
differential equations in a space of matrices and derive from it a matrix
Riccati hierarchy. The latter is then shown to exhibit an underlying 'weakly
nonassociative' (WNA) algebra structure, from which we can conclude, refering
to previous work, that any solution of the Riccati system also solves the
potential KP hierarchy (in the corresponding matrix algebra). We then turn to
the case where the components of the matrices are multiplied using a
(generalized) star product. Associated with the deformation parameters, there
are additional symmetries (flow equations) which enlarge the respective KP
hierarchy. They have a compact formulation in terms of the WNA structure. We
also present a formulation of the KP hierarchy equations themselves as
deformation flow equations.Comment: 25 page
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
All bicovariant differential calculi on Glq(3,C) and SLq(3,C)
All bicovariant first order differential calculi on the quantum group
GLq(3,C) are determined. There are two distinct one-parameter families of
calculi. In terms of a suitable basis of 1-forms the commutation relations can
be expressed with the help of the R-matrix of GLq(3,C). Some calculi induce
bicovariant differential calculi on SLq(3,C) and on real forms of GLq(3,C). For
generic deformation parameter q there are six calculi on SLq(3,C), on SUq(3)
there are only two. The classical limit q-->1 of bicovariant calculi on
SLq(3,C) is not the ordinary calculus on SL(3,C). One obtains a deformation of
it which involves the Cartan-Killing metric.Comment: 24 pages, LaTe
Noncommutative Geometry of Finite Groups
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).Comment: 34 pages, Late
Differential Geometry of Group Lattices
In a series of publications we developed "differential geometry" on discrete
sets based on concepts of noncommutative geometry. In particular, it turned out
that first order differential calculi (over the algebra of functions) on a
discrete set are in bijective correspondence with digraph structures where the
vertices are given by the elements of the set. A particular class of digraphs
are Cayley graphs, also known as group lattices. They are determined by a
discrete group G and a finite subset S. There is a distinguished subclass of
"bicovariant" Cayley graphs with the property that ad(S)S is contained in S.
We explore the properties of differential calculi which arise from Cayley
graphs via the above correspondence. The first order calculi extend to higher
orders and then allow to introduce further differential geometric structures.
Furthermore, we explore the properties of "discrete" vector fields which
describe deterministic flows on group lattices. A Lie derivative with respect
to a discrete vector field and an inner product with forms is defined. The
Lie-Cartan identity then holds on all forms for a certain subclass of discrete
vector fields.
We develop elements of gauge theory and construct an analogue of the lattice
gauge theory (Yang-Mills) action on an arbitrary group lattice. Also linear
connections are considered and a simple geometric interpretation of the torsion
is established.
By taking a quotient with respect to some subgroup of the discrete group,
generalized differential calculi associated with so-called Schreier diagrams
are obtained.Comment: 51 pages, 11 figure
Bicomplexes and Integrable Models
We associate bicomplexes with several integrable models in such a way that
conserved currents are obtained by a simple iterative construction. Gauge
transformations and dressings are discussed in this framework and several
examples are presented, including the nonlinear Schrodinger and sine-Gordon
equations, and some discrete models.Comment: 17 pages, LaTeX, uses amssymb.sty and diagrams.st