Erlangen Program at Large-1: Geometry of Invariants

Kisil, Vladimir V.

doi:10.3842/sigma.2010.076

Cited by 17 publications

(38 citation statements)

References 52 publications

(77 reference statements)

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“…The design of the library figure shaped the general theoretical approach to the extension of Möbius-Lie geometry [2,3], which leaded to specific realisations in [33,34,35]. Furthermore, it shall be helpful for computer experiments in Lie sphere geometry of indefinite or nilpotent metrics since our intuition is not elaborated there in contrast to the Euclidean space [22,26,36]. Some advances in the two-dimensional space were achieved recently [13,37], however further developments in higher dimensions are still awaiting their researchers.…”

Section: Discussionmentioning

confidence: 99%

“…More specifically, the first library cycle [1,13,22] manipulates individual cycles within the GiNaC [23] computer algebra system. The mathematical formalism employed in the library cycle is based on Clifford algebras and the Fillmore-Springer-Cnops construction (FSCc), which has a long history, see [24, § 1.1], [20, § 4.1], [19], [25, § 4.2], [26], [13, § 4.2]. Compared to a plain analytical treatment [27,11], FSCc is much more efficient and conceptually coherent in dealing with FLT-invariant properties of cycles.…”

Section: Problems and Backgroundmentioning

confidence: 99%

“…The first library cycle was initially reported in [1] and since then was employed as the main research tool in several papers [26,22] and monograph [13]. New results, e.g.…”

Section: Software Impactmentioning

confidence: 99%

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MoebInv: C++ libraries for manipulations in non-Euclidean geometry

Kisil¹

2020

SoftwareX

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The introduced package MoebInv contains two C ++ libraries for symbolic, numeric and graphical manipulations in non-Euclidean geometry. The first library cycle implements basic geometric operations on cycles, which are the zero sets of certain polynomials of degree two. The second library figure operates on ensembles of cycles interconnected by Moebius-invariant relations: orthogonality, tangency, etc. Both libraries work in spaces with any dimension and arbitrary signatures of their metrics. Their essential functionality is accessible in interactive modes from Python/Jupyter shells and a dedicated Graphical User Interface. The latter does not require any coding skills and can be used in education. The package is tested on (and supplied for) various Linux distributions, Windows 10, Mac OS X and several cloud services.

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Section: Discussionmentioning

confidence: 99%

Section: Problems and Backgroundmentioning

confidence: 99%

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MoebInv: C++ libraries for manipulations in non-Euclidean geometry

Kisil¹

2020

SoftwareX

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“…The equivalent identity T J = JT −1 produces a system of homogeneous linear equations which has the generic solution: 44 c with four free variables j 11 , j 42 , j 43 and j 44 . Since a solution shall not depend on a, b, c, d, we have to put j 42 = j 44 = 0.…”

Section: Extending Cyclesmentioning

confidence: 99%

Möbius--Lie Geometry and Its Extension

Kisil¹

2019

GIQ

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These lectures review the classical Möbius-Lie geometry and recent work on its extension. The latter considers ensembles of cycles (quadrics), which are interconnected through conformal-invariant geometric relations (e.g. "to be orthogonal", "to be tangent", etc.), as new objects in an extended Möbius-Lie geometry. It is shown on examples, that such ensembles of cycles naturally parameterise many other conformally-invariant families of objects, two examples-the Poincaré extension and continued fractions are considered in detail. Further examples, e.g. loxodromes, wave fronts and integrable systems, are published elsewhere.The extended Möbius-Lie geometry is efficient due to a method, which reduces a collection of conformally invariant geometric relations to a system of linear equations, which may be accompanied by one fixed quadratic relation. The algorithmic nature of the method allows to implement it as a C++ library, which operates with numeric and symbolic data of cycles in spaces of arbitrary dimensionality and metrics with any signatures. Numeric calculations can be done in exact or approximate arithmetic. In the two-and three-dimensional cases illustrations and animations can be produced. An interactive Python wrapper of the library is provided as well.

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“…Notably the action (21) is a group homomorphism of the group SL 2 (R) into transformations of the "upper half-plane" on hypercomplex numbers. Although dual and double numbers are algebraically trivial, the respective geometries in the spirit of Erlangen programme are refreshingly inspiring [50,66,70] and provide useful insights even in the elliptic case [61]. In order to treat divisors of zero, we need to consider Möbius transformations (21) of conformally completed plane [34,62].…”

Section: Groups Homogeneous Spaces and Hypercomplex Numbersmentioning

confidence: 99%

Symmetry, Geometry and Quantization with Hypercomplex Numbers

Kisil¹

2017

GIQ

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These notes describe some links between the group SL 2 (R), the Heisenberg group and hypercomplex numbers -complex, dual and double numbers. Relations between quantum and classical mechanics are clarified in this framework. In particular, classical mechanics can be obtained as a theory with noncommutative observables and a non-zero Planck constant if we replace complex numbers in quantum mechanics by dual numbers. Our consideration is based on induced representations which are build from complex-/dual-/double-valued characters. Dynamic equations, rules of additions of probabilities, ladder operators and uncertainty relations are discussed. Finally, we prove a Calderón-Vaillancourt-type norm estimation for relative convolutions.

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Erlangen Program at Large-1: Geometry of Invariants

Cited by 17 publications

References 52 publications

MoebInv: C++ libraries for manipulations in non-Euclidean geometry

MoebInv: C++ libraries for manipulations in non-Euclidean geometry

Möbius--Lie Geometry and Its Extension

Symmetry, Geometry and Quantization with Hypercomplex Numbers

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