Derivation based differential calculi for noncommutative algebras deforming a class of three dimensional spaces

Marmo, Giuseppe; Vitale, Patrizia; Zampini, Alessandro

doi:10.1016/j.geomphys.2018.10.013

Cited by 9 publications

(7 citation statements)

References 51 publications

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“…with k a suitable dimensionful constant, but the limit, Θ → 0, does not yield the correct commutative limit (see [13][14][15][16][17] for details and applications). A related approach, is to use twisted differential calculus for those NC algebras whose star product is defined in terms of a twist [18][19][20][21][22].…”

Section: Jhep08(2020)041mentioning

confidence: 99%

A novel approach to non-commutative gauge theory

Kupriyanov

Vitale

2020

J. High Energ. Phys.

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We propose a field theoretical model defined on non-commutative space-time with non-constant non-commutativity parameter Θ(x), which satisfies two main requirements: it is gauge invariant and reproduces in the commutative limit, Θ → 0, the standard U(1) gauge theory. We work in the slowly varying field approximation where higher derivatives terms in the star commutator are neglected and the latter is approximated by the Poisson bracket, −i[f, g] ≈ {f, g}. We derive an explicit expression for both the NC deformation of Abelian gauge transformations which close the algebra [δ f , δ g ]A = δ {f,g} A, and the NC field strength F , covariant under these transformations, δ f F = {F , f }. NC Chern-Simons equations are equivalent to the requirement that the NC field strength, F , should vanish identically. Such equations are non-Lagrangian. The NC deformation of Yang-Mills theory is obtained from the gauge invariant action, S = F 2. As guiding example, the case of su(2)-like non-commutativity, corresponding to rotationally invariant NC space, is worked out in detail.

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Section: Jhep08(2020)041mentioning

confidence: 99%

A novel approach to non-commutative gauge theory

Kupriyanov

Vitale

2020

J. High Energ. Phys.

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“…while Y = u 2 X3 − u 3 X2 and then µ * (Y ) = 0. These examples have been considered in relation with a Lie-Poisson dynamics and in T-duality field theory, see [27,28], while the properties of the associated momentum map have been used to study non commutative differential calculi in [34,35].…”

Section: 32mentioning

confidence: 99%

“…so one has the graded 35 exterior algebra (Λ(M ) = ⊕ N k=0 Λ k (M ), ∧), with Λ 0 (M ) = F(M ) and elements having the local representation…”

Section: 2mentioning

confidence: 99%

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Symmetries and Reduction -- part I -- Poisson and symplectic picture

Marmo,

Schiavone,

Zampini

2020

Preprint

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Coherently with the principle of analogy suggested by Dirac, we describe a general setting for reducing a classical dynamics, and the role of the Noether theorem -connecting symmetries with constants of the motion -within a reduction. This is the first of two papers, and it focuses on the reduction within the Poisson and the symplectic formalism. 1 4.2. Exact symplectic manifolds 23 4.3. Symmetries and reduction within the symplectic formalism 24 4.4. Symmetries for linear symplectic dynamics 36 5. Symmetries and conservation laws for presymplectic dynamics 40 5.1. A Noether theorem for global dynamics on pre-symplectic systems 42 5.2. Symmetries and constants of the motion for pre-symplectic systems 43 Appendix A. The exterior differential calculus on a manifold 44 A.1. Cartan calculus on a manifold 44 A.2. Tangent maps and Pull-backs 47 Appendix B. Distributions and foliations. The Frobenius theorem 47 References 49

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“…We limit ourselves to notice that the splitting we considered in the classical setting depends on the dynamics and therefore implicitly assumes the existence of a differentiable (smooth, indeed) structure on M . We turn now our attention to describe how it is possible to define a derivation based differential calculus on an algebra A (see [8,23,24,27,28]). The set Λ k (A) of Z(A)-multilinear alternating maps (with X j ∈ Der(A))…”

Section: 1mentioning

confidence: 99%

Abstract Dynamical Systems: Remarks on Symmetries and Reduction

Marmo,

Zampini

2020

Preprint

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We review how an algebraic formulation for the dynamics of a physical system allows to describe a reduction procedure for both classical and quantum evolutions. Contents1. An introduction Acknowledgments 2. An algebraic description of a dynamical system 2.1. Observables and states 2.2. Derivations and dynamics 2.3. Quantum and classical dynamics 2.4. Classical and quantum integrability 2.5. About the Wigner's problem 3. Symmetries and reduction for a classical dynamics 4. A reduction scheme for quantum dynamics 4.1. A differential calculus on a non commutative space 5. Conclusions References

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Derivation based differential calculi for noncommutative algebras deforming a class of three dimensional spaces

Abstract: We equip a family of algebras whose noncommutativity is of Lie type with a derivation based differential calculus obtained, upon suitably using both inner and outer derivations, as a reduction of a redundant calculus over the Moyal four dimensional space. 2

Cited by 9 publications

References 51 publications

A novel approach to non-commutative gauge theory

A novel approach to non-commutative gauge theory

Symmetries and Reduction -- part I -- Poisson and symplectic picture

Abstract Dynamical Systems: Remarks on Symmetries and Reduction

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