Group conjugations of Dirac operators as an invariant of the Riemannian manifold

Klishevich, V. V.

doi:10.1088/0264-9381/19/16/305

Cited by 3 publications

(30 citation statements)

References 13 publications

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“…We know that in many particular cases [9,8,6] this is possible and now we intend to point out that this is a general property of theories involving roots. To this end we introduce a new useful point-dependent matrix.…”

Section: Continuous Symmetries Generated By Unit Rootsmentioning

confidence: 89%

“…There are two families of three roots [8]. The unit roots of the first triplet, F , have the non-vanishing complex components [8] f The discrete symmetry is given by two representations of the quaternion group acting on each of both chiral sectors. On the left-handed sector acts the group Q(F ) represented by the operators I, P = diag(−1 2 , 1 2 ), U (i) = diag(−iσ i , 1 2 ) and P U (i) .…”

Section: Appendix C Example: Minkowski Spacetimementioning

confidence: 99%

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Symmetries of the Dirac operators associated with covariantly constant Killing–Yano tensors

Cotăescu

Visinescu

2003

Class. Quantum Grav.

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The continuous and discrete symmetries of the Dirac-type operators produced by particular Killing-Yano tensors are studied in manifolds of arbitrary dimensions. The Killing-Yano tensors considered are covariantly constant and realize certain square roots of the metric tensor. Such a Killing-Yano tensor produces simultaneously a Dirac-type operator and the generator of a one-parameter Lie group connecting this operator with the standard Dirac one. The Dirac operators are related among themselves through continuous or discrete transformations. It is shown that the groups of the continuous symmetry can be only U (1) and SU (2), specific to (hyper-)Kähler spaces, but arising even in cases when the requirements for these special geometries are not fulfilled. The discrete symmetries are also studied obtaining the discrete groups Z 4 and Q. The briefly presented examples are the Euclidean Taub-NUT space and the Minkowski spacetime.Pacs 04.62.+v

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Section: Continuous Symmetries Generated By Unit Rootsmentioning

confidence: 89%

Section: Appendix C Example: Minkowski Spacetimementioning

confidence: 99%

Symmetries of the Dirac operators associated with covariantly constant Killing–Yano tensors

Cotăescu

Visinescu

2003

Class. Quantum Grav.

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“…The Minkowski spacetime possesses a pair of adjoint triplets f = f * [10]. The unit roots of the first triplet, f = {f 1 , f 2 , f 3 }, have the non-vanishing complex-valued components [10] …”

Section: Dirac-type Operatorsmentioning

confidence: 99%

“…The unit roots of the first triplet, f = {f 1 , f 2 , f 3 }, have the non-vanishing complex-valued components [10] …”

Section: Dirac-type Operatorsmentioning

confidence: 99%

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Superalgebras of Dirac operators on manifolds with special Killing‐Yano tensors

Cotăescu

Visinescu²

2006

Fortschritte der Physik

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We present the properties of new Dirac-type operators generated by real or complex-valued special Killing-Yano tensors that are covariantly constant and represent roots of the metric tensor. In the real case these are just the so called complex or hyper-complex structures of the Kählerian manifolds. Such a Killing-Yano tensor produces simultaneously a Dirac-type operator and the generator of a one-parameter Lie group connecting this operator with the standard Dirac one. In this way the Dirac operators are related among themselves through continuous transformations associated with specific discrete ones. We show that the group of these continuous transformations can be only U (1) or SU (2). It is pointed out that the Dirac and Dirac-type operators can form N = 4 superalgebras whose automorphisms combine isometries with the SU (2) transformation generated by the Killing-Yano tensors. As an example we study the automorphisms of the superalgebras of Dirac operators on Minkowski spacetime.Pacs 04.62.+v *

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Symmetries and Supersymmetries of the Dirac-Type Operators on Euclidean Taub-Nut Space

Cotăescu¹,

Visinescu²

2005

Mathematical, Theoretical and Phenomenological Challenges Beyond the Standard Model

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It is shown that the main geometrical objects involved in all the symmetries or supersymmetries of the Dirac operators in curved manifolds of arbitrary dimensions are the Killing vectors and the Killing-Yano tensors of any ranks. The general theory of external symmetry transformations associated to the usual isometries is presented, pointing out that these leave the standard Dirac equation invariant providing the correct spin parts of the group generators. Furthermore, one analyzes the new type of symmetries generated by the covariantly constant Killing-Yano tensors that realize certain square roots of the metric tensor. Such a Killing-Yano tensor produces simultaneously a Dirac-type operator and the generator of a one-parameter Lie group connecting this operator with the standard Dirac one. In this way the Dirac operators are related among themselves through continuous transformations associated to specific discrete ones. It is shown that the groups of this continuous symmetry can be only U (1) or SU (2), as those of the (hyper-)Kähler spaces, but arising even in cases when the requirements for these special geometries are not fulfilled. Arguments are given that for the non-Kählerian manifolds it is convenient to enlarge this SU (2) symmetry up to a SL(2, C) one through complexification. In other respects, it is pointed out that the Dirac-type operators can form N = 4 superalgebras whose automorphisms combine external symmetry transformations with those of the mentioned SU (2) or SL(2, C) groups. The discrete symmetries are also studied obtaining the discrete groups Z 4 and Q. To exemplify, the Euclidean Taub-NUT space with its Dirac-type operators is presented in much details, pointing out that there is an infinite-loop superalgebra playing the role of a * Final version of Ref.[1] containing new results, extended discussions and some corrections. † closed dynamical algebraic structure. As a final topic, we go to consider the properties of the Dirac-type operators of the Minkowski spacetime.Pacs 04.62.+v

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Group conjugations of Dirac operators as an invariant of the Riemannian manifold

Cited by 3 publications

References 13 publications

Symmetries of the Dirac operators associated with covariantly constant Killing–Yano tensors

Symmetries of the Dirac operators associated with covariantly constant Killing–Yano tensors

Superalgebras of Dirac operators on manifolds with special Killing‐Yano tensors

Symmetries and Supersymmetries of the Dirac-Type Operators on Euclidean Taub-Nut Space

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