Possible contractions of quantum orthogonal groups

Gromov, Nikolay; Kostyakov, I. V.; Kuratov, V. V.

doi:10.1134/1.1432910

Cited by 8 publications

(12 citation statements)

References 23 publications

(46 reference statements)

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“…It deemed to be unavoidable [46, § 3.3.4] to employ complex, dual and double numbers to represent three different types of Möbius transformations extended from the real line to a plane. Also (hyper)complex numbers were essential in [40,46] to define three possible types of cycle product (30), and now we managed without them.…”

Section: Summary Of the Construction And Generalisationsmentioning

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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Section: Summary Of the Construction And Generalisationsmentioning

confidence: 99%

Möbius--Lie Geometry and Its Extension

Kisil¹

2019

GIQ

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“…[14] (another definition of nilpotent numbers can be found in Ref. [9]). Let us consider simple example of one variable case.…”

Section: Nilpotent Commuting Variablesmentioning

confidence: 99%

“…Numbers with nilpotent part are known in mathematics for a long time. Generalized dual numbers were used by N. A. Gromov [9,10,11] in a series of papers devoted to the contractions of groups, quantum groups and description of spaces with degenerate metrics and relevant field theories. Earlier P. I. Pimenov [12] has given classification of spaces of the constant curvature using nilpotent commuting numbers.…”

Section: Introductionmentioning

confidence: 99%

Nilpotent fuzz

Frydryszak¹

2008

Reports on Mathematical Physics

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“…[11,12]. It is worth noting that Gromov, in a series of papers, applied dual numbers in several ways: in contractions and analytical continuations of classical groups [13], then in quantum group formalism [14,15].…”

Section: Introductionmentioning

confidence: 99%