• Corpus ID: 18044584

Deformations of Classical Geometries and Integrable Systems

@article{Dimakis1997DeformationsOC,
  title={Deformations of Classical Geometries and Integrable Systems},
  author={Aristophanes Dimakis and Folkert Muller-Hoissen},
  journal={arXiv: Mathematical Physics},
  year={1997}
}
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. 
3 Citations

Discrete Riemannian geometry

Within a framework of noncommutative geometry, we develop an analog of (pseudo-) Riemannian geometry on finite and discrete sets. On a finite set, there is a counterpart of the continuum metric

The generalized Moyal–Nahm and continuous Moyal–Toda equations

We present in detail a class of solutions to the 4D SU(∞) Moyal–anti-self-dual Yang–Mills (ASDYM) equations (an effective 6D theory) that are related to reductions of the generalized Moyal–Nahm

Obituary: Aristophanes Dimakis

The theoretical physicist and mathematician Aristophanes Dimakis passed away on July 8, 2021, at the age of 68, in Athens, Greece. We briefly review his life, career and scientific achievements. We

References

SHOWING 1-10 OF 18 REFERENCES

Noncommutative Geometry and Integrable Models

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 theordinary calculus

Quantum mechanics as noncommutative symplectic geometry

The authors construct consistent differential calculi on the algebra generated by operators satisfying the canonical commutation relation. This leads to a mathematical framework in which quantum

Some aspects of noncommutative geometry and physics

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

Noncommutative differential calculus and lattice gauge theory

The authors study consistent deformations of the classical differential calculus on algebras of functions (and, more generally, commutative algebras) such that differentials and functions satisfy

Noncommutative Geometry

Noncommutative Spaces It was noticed a long time ago that various properties of sets of points can be restated in terms of properties of certain commutative rings of functions over those sets. In

Compact matrix pseudogroups

The compact matrix pseudogroup is a non-commutative compact space endowed with a group structure. The precise definition is given and a number of examples is presented. Among them we have compact

Low-dimensional Sigma Models,

Introduction to path integrals - quantum mechanics fermions functional integrals in field theory further comments on functional methods green functions in field theory application of the saddly point

The consistent reduction of the differential calculus on the quantum group $GL_{q}(2,C)$ to the differential calculi on its subgroups and $\sigma$-models on the quantum group manifolds $SL_{q}(2,R)$, $SL_{q}(2,R)/U_{h}(1)$, $C{q}(2|0)$ and infinitesimal transformations

Explicit construction of the second order left differential calculi on the quantum group and its subgroups are obtained with the property of the natural reduction: the differential calculus on the