A new application of k-symplectic Lie systems

Lucas, J. de; Tobolski, Mariusz; Vilariño, Silvia

doi:10.1142/s0219887815500711

Cited by 2 publications

(4 citation statements)

References 8 publications

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“…However, it is noted in [15] that there is not in general a natural associative product on C ∞ H (M, V) with respect to which {, } would be a Poisson bracket.…”

Section: Now For Every G ∈ Gmentioning

confidence: 99%

“…, ω k ) on M consists of k closed 2-forms ω i with ∩ i ker ω i = 0. Indeed, the terms k-symplectic and polysymplectic appear nearly interchangeably throughout the literature, though in the latter case a condition that k + 1 divide dim M is sometimes imposed [15]. We also note the independent introduction of this material as the k-almost cotangent formalism in [12,13].…”

Section: Introductionmentioning

confidence: 99%

“…In addition to its applications in classical field theory [19,40,43,48,50], polysymplectic geometry is also the subject of intrinsic mathematical interest [7,10,12,52]. In this regard, we note the independent work of Norris [45][46][47] on the canonical polysymplectic structure of the frame bundle on a smooth manifold, as well as the more recent appearance of k-symplectic Lie systems [15,16]. Additionally, steps have been taken to relate polysymplectic geometry to the related study of multisymplectic geometry [17], which similarly arises in field-theoretic contexts [11,38,44].…”

Section: Introductionmentioning

confidence: 99%

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Polysymplectic reduction and the moduli space of flat connections

Blacker¹

2019

J. Phys. A: Math. Theor.

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A polysymplectic structure is a vector-valued symplectic form, that is, a closed nondegenerate 2-form with values in a vector space. We first outline the polysymplectic Hamiltonian formalism with coefficients in a vector space V , we then apply this framework to show that the moduli space of flat connections on a principal bundle over a compact manifold M is a polysymplectic reduction of the space of all connections by the action of the gauge group with respect to a natural polysymplectic structure with values in an infinite dimensional Banach space. As a consequence, the moduli space inherits a canonical H 2 (M )-valued presymplectic structure.Along the way, we establish various properties of polysymplectic manifolds. For example, a Darboux-type theorem asserts that every V -symplectic manifold locally symplectically embeds in a standard polysymplectic manifold Hom(T Q, V ). We also show that both the Arnold conjecture and the well-known convexity properties of the classical moment map fail to hold in the polysymplectic setting.

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“…However, it is noted in [15] that there is not in general a natural associative product on C ∞ H (M, V) with respect to which {, } would be a Poisson bracket.…”

Section: Now For Every G ∈ Gmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

Polysymplectic reduction and the moduli space of flat connections

Blacker¹

2019

J. Phys. A: Math. Theor.

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“…It is worth noting that geometric techniques, e.g. symplectic, Poisson, k-symplectic or Jacobi structures, may be applied to study their properties [7,22,23,30].…”

Section: Introductionmentioning

confidence: 99%

Geometry of Riccati equations over normed division algebras

Lucas

Tobolski

Vilariño

2016

Journal of Mathematical Analysis and Applications

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This work presents and studies Riccati equations over finite-dimensional normed division algebras. We prove that a Riccati equation over a finite-dimensional normed division algebra A is a particular case of conformal Riccati equation on an Euclidean space and it can be considered as a curve in a Lie algebra of vector fields V so(dim A + 1, 1). Previous results on known types of Riccati equations are recovered from a new viewpoint. A new type of Riccati equations, the octonionic Riccati equations, are extended to the octonionic projective line OP 1 . As a new physical application, quaternionic Riccati equations are applied to study quaternionic Schrödinger equations on 1+1 dimensions.

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A new application of k-symplectic Lie systems

Cited by 2 publications

References 8 publications

Polysymplectic reduction and the moduli space of flat connections

Polysymplectic reduction and the moduli space of flat connections

Geometry of Riccati equations over normed division algebras

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