The Gel’fand–Tsetlin representations of the orthogonal Cayley–Klein algebras

Gromov, Nikolay

doi:10.1063/1.529711

Cited by 5 publications

(1 citation statement)

References 18 publications

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“…Sanjuan constructs the Lie algebra for the hyperbolic plane using the standard method, stating that this method can be used to obtain the other Lie algebras as well. Also, extensive work has been done by Gromov [6,7,8,9, 10] on the generalized orthogonal groups SO(3) (which we refer to simply as SO(3)), deriving representations of the generalized so(3) (which we refer to simply as so (3)) by utilizing the dual numbers as well as the standard complex numbers, where again it is tacitly assumed that the parameters κ 1 and κ 2 have been normalized. Also, Pimenov has given an axiomatic description of all Cayley-Klein spaces in arbitrary dimensions in his paper [22] via the dual numbers i k , k = 1, 2, .…”

Section: Cayley-klein Geometriesmentioning

confidence: 99%

Clifford Algebras and Possible Kinematics

McRae¹

2007

SIGMA

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Abstract. We review Bacry and Lévy-Leblond's work on possible kinematics as applied to 2-dimensional spacetimes, as well as the nine types of 2-dimensional Cayley-Klein geometries, illustrating how the Cayley-Klein geometries give homogeneous spacetimes for all but one of the kinematical groups. We then construct a two-parameter family of Clifford algebras that give a unified framework for representing both the Lie algebras as well as the kinematical groups, showing that these groups are true rotation groups. In addition we give conformal models for these spacetimes. The first part of this paper is a review of Bacry and Lévy-Leblond's description of possible kinematics and how such kinematical structures relate to the Cayley-Klein formalism. We review some of the work done by Ballesteros, Herranz, Ortega and Santander on homogeneous spaces, as this work gives a unified and detailed description of possible kinematics (save for static kinematics). The second part builds on this work by analyzing the corresponding kinematical models from other unified viewpoints, first through generalized complex matrix realizations and then through a two-parameter family of Clifford algebras. These parameters are the same as those given by Ballesteros et. al., and relate to the speed of light and the universe time radius. Part I. A review of kinematics via Cayley-Klein geometries 1 Possible kinematicsAs noted by Inonu and Wigner in their work [17] on contractions of groups and their representations, classical mechanics is a limiting case of relativistic mechanics, for both the Galilei group as well as its Lie algebra are limits of the Poincaré group and its Lie algebra. Bacry and Lévy-Leblond [1] classified and investigated the nature of all possible Lie algebras for kinematical groups (these groups are assumed to be Lie groups as 4-dimensional spacetime is assumed to be continuous) given the three basic principles that (i) space is isotropic and spacetime is homogeneous,(ii) parity and time-reversal are automorphisms of the kinematical group, and (iii) the one-dimensional subgroups generated by the boosts are non-compact.

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Section: Cayley-klein Geometriesmentioning

confidence: 99%

Clifford Algebras and Possible Kinematics

McRae¹

2007

SIGMA

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Thermodynamic properties of quantum models based on the complex unitary Cayley-Klein groups

Najafizade

Panahi

Hassanabadi

2020

Physica A: Statistical Mechanics and its Applications

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SO(5)q and contraction: Chevalley basis representations for q-generic and root of unity

Chakrabarti

1994

Journal of Mathematical Physics

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November 1993 OCR Output A270.ll93 for the non-simisimple contracted case, for all q.representations have their own interest. Moreover they lead to a consistent Hopf algebra and representations contracting the generator J 45 in the G-Z formalism. The contracted representations are given.The SO (5) is quite different, even at the classical level, from the familiar E(4), the 4 dim. Eucl. algebra, obtained on generators (e2, f2) leads to an easily obtained complete Hopf algebra. The structure of the contracted algebra restriction we show how to construct all representations for all q's. The contraction of the Chevalley irreducible representations labelled by 3 variable parameters, the maximal number being 4. Within this elsewhere, in q-deformations for orthogonal (though not for unitary) algebras. Our results are limited to sharp contrast to the classical canonical Gelfand-Zetlin representations which lead to problems, studied once obtained on this basis, are very simply deformed for arbitrary q, generic and root of unity. This is in Representations of SO (5) 'fractional part" discussed in detail in Sec.4. In fact, for q a root of unity, we have already presented OCR Output has been possible to unqfy the presentation including the root of unity case thanks to our formalism of chosen to present the representations directly in a form valid for all q (q=l, q¢1 and real, q = e). It *2"'N constant factors of (q+q·l)as in (2.13), (2.15) and (2.16). The q-deformation being simple, we have q-brackets defined in (2.1) along with [x] due to the existence of unequal roots. There are also external easy. The differences with the "minimal" q-deformation for the unitary case are in the appearences of [x]2Having constructed the classical solutions, the q-deformation, as anticipated, turned out to be the Appendix leading to (A.l5) and what follows. construct explicitly. But awareness of this relation, even if implicit, can be helpful. See the discussion in exhibit the equivalence of the GZ with our representations. The matrix diagonalizing J45 is too difficult to correspondance of the canonical and the Chevalley bases, the G Z generator J45 has to be diagonalized to detail in [8,9]. In our conventions, which we found to be the most convenient concerning the diagonalized. The problems due to a non-diagonalized Cartan generator in q-deformation is studied in particularly crucial since from hi we have to go over to qihi for q¢l. In the GZ formalism only hi = Ji2 is subset is diagonalized. Both these features are absent for SO(n). The question of diagonalisation is generators necessary for building the whole algebra, is thus the same in each formalism and the same hi to -the latter being already diagonalized in the GZ representations. The basic, minimal set of ' generators (eifi) correspond to (A'+f, Ai+i) of the canonical generators directly and the Cartan generators representations had to be constructed. For unitary algebras this problem does not exist. The Chevalley minimally on nearly so. This hopeful conjecture turned out to b...

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The Gel’fand–Tsetlin representations of the orthogonal Cayley–Klein algebras

Cited by 5 publications

References 18 publications

Clifford Algebras and Possible Kinematics

Clifford Algebras and Possible Kinematics

Thermodynamic properties of quantum models based on the complex unitary Cayley-Klein groups

SO(5)q and contraction: Chevalley basis representations for q-generic and root of unity

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