96 research outputs found
Quasi-symmetric functions and the KP hierarchy
Quasi-symmetric functions show up in an approach to solve the
Kadomtsev-Petviashvili (KP) hierarchy. This moreover features a new
nonassociative product of quasi-symmetric functions that satisfies simple
relations with the ordinary product and the outer coproduct. In particular,
supplied with this new product and the outer coproduct, the algebra of
quasi-symmetric functions becomes an infinitesimal bialgebra. Using these
results we derive a sequence of identities in the algebra of quasi-symmetric
functions that are in formal correspondence with the equations of the KP
hierarchy.Comment: 16 page
Pythagoras' Theorem on a 2D-Lattice from a "Natural" Dirac Operator and Connes' Distance Formula
One of the key ingredients of A. Connes' noncommutative geometry is a
generalized Dirac operator which induces a metric(Connes' distance) on the
state space. We generalize such a Dirac operator devised by A. Dimakis et al,
whose Connes' distance recovers the linear distance on a 1D lattice, into 2D
lattice. This Dirac operator being "naturally" defined has the so-called "local
eigenvalue property" and induces Euclidean distance on this 2D lattice. This
kind of Dirac operator can be generalized into any higher dimensional lattices.Comment: Latex 11pages, no figure
Differential Calculi on Quantum Spaces determined by Automorphisms
If the bimodule of 1-forms of a differential calculus over an associative
algebra is the direct sum of 1-dimensional bimodules, a relation with
automorphisms of the algebra shows up. This happens for some familiar quantum
space calculi.Comment: 7 pages, Proceedings of XIIIth International Colloquium Integrable
Systems and Quantum Group
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.Comment: Latex; 13 pages; no figures. References added. Accepted by Phys.
Lett.
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