Graded Poisson structures on the algebra of differential forms

Beltrán, José; Monterde, J.

doi:10.1007/bf02566014

Cited by 17 publications

(17 citation statements)

References 8 publications

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“…In this case it is clear from the formula on higher degrees that L δ (ω, η) corresponds to the Schouten bracket of alternating multivector fields cf. [6], and indeed the results of Lemma 2.4 can be seen as parallel to properties of this [5,10]. There will also be a form of graded Jacobi identity which we have not elaborated here as we will not need it.…”

Section: Reconstruction From a Codifferentialmentioning

confidence: 88%

Reconstruction and quantization of Riemannian structures

Majid

2020

Journal of Mathematical Physics

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We study how the Riemannian structure on a manifold can be usefully reconstructed from its codifferential δ, including a formula ∇ωη = 1 2 (δ(ωη) − (δω)η + ω(δη) + Lω η + iηdω) for the Levi-Civita covariant derivative in terms of 1-forms, where L, i are respectively the Lie derivative and interior product along the corresponding vector fields. The covariant derivative extends naturally along forms of any degree and to possibly degenerate ( , ). In the nondegenerate case, δ makes the exterior algebra into a BV algebra. In the invertible case we show that Ricci = − 1 2 ∆g where the Hodge Laplacian ∆ extends in a natural way to act on the metric.Our results come from a new way of thinking about metrics and connections as a kind of cocycle data for central extensions of differential graded algebras (DGAs), a theory which we introduce. We show that any cleft extension of the exterior algebra Ω(M ) on a manifold is associated to a possiblydegenerate metric and covariant derivative. Those for which d is not deformed up to isomorphism correspond to the Levi-Civita case. We provide a construction for such extensions both of classical DGAs and of already nongraded-commutative DGAs, thereby constructing a class of bimodule covariant derivatives via a kind of quantum analogue of the Koszul formula.We also provide a semidirect product of any differential graded algebra by the quantum differential algebra Ω(t, dt) in one variable, to introduce a noncommutative 'time'. Composing these two constructions recovers a differential quantisation of M × R in [12].

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Section: Reconstruction From a Codifferentialmentioning

confidence: 88%

Reconstruction and quantization of Riemannian structures

Majid

2020

Journal of Mathematical Physics

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“…By (6) In terms of the differentials 6, and 6j , j , Equation (8) [2], [3], followed by, among others, [43], [17], [37], [18], [1], or of string theories [11]. See also [10] and [39] [41] and [37], formula (20), a fact already observed by Witten in [43], formula (13 volume element /-t on M such that IIT*M, with the Schouten bracket, the odd vector field dp, and the berezinian volume çJ-L' is a QSP-manifold.…”

mentioning

confidence: 99%

Divergence operators and odd Poisson brackets

Kosmann–Schwarzbach¹,

Monterde²

2002

Annales De L’institut Fourier

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We define the divergence operators on a graded algebra, and we show that, given an odd Poisson bracket on the algebra, the operator that maps an element to the divergence of the hamiltonian derivation that it defines is a generator of the bracket. This is the "odd laplacian", ∆, of Batalin-Vilkovisky quantization. We then study the generators of odd Poisson brackets on supermanifolds, where divergences of graded vector fields can be defined either in terms of berezinian volumes or of graded connections. Examples include generators of the Schouten bracket of multivectors on a manifold (the supermanifold being the cotangent bundle where the coordinates in the fibres are odd) and generators of the Koszul-Schouten bracket of forms on a Poisson manifold (the supermanifold being the tangent bundle, with odd coordinates on the fibres).

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“…In this section, examples of DG Poisson algebras are provided arising from Lie theory, differential geometry, homological algebra and deformation theory. See more examples in [5,9]. 5.1.…”

Section: Further Examples Of Dg Poisson Algebrasmentioning

confidence: 99%

DG Poisson algebra and its universal enveloping algebra

Lü¹,

Wang²,

Zhuang³

2016

Sci. China Math.

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In this paper, we introduce the notions of differential graded (DG) Poisson algebra and DG Poisson module. Let $A$ be any DG Poisson algebra. We construct the universal enveloping algebra of $A$ explicitly, which is denoted by $A^{ue}$. We show that $A^{ue}$ has a natural DG algebra structure and it satisfies certain universal property. As a consequence of the universal property, it is proved that the category of DG Poisson modules over $A$ is isomorphic to the category of DG modules over $A^{ue}$. Furthermore, we prove that the notion of universal enveloping algebra $A^{ue}$ is well-behaved under opposite algebra and tensor product of DG Poisson algebras. Practical examples of DG Poisson algebras are given throughout the paper including those arising from differential geometry and homological algebra.Comment: Accepted by Science China Mathematic

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Graded Poisson structures on the algebra of differential forms

Cited by 17 publications

References 8 publications

Reconstruction and quantization of Riemannian structures

Reconstruction and quantization of Riemannian structures

Divergence operators and odd Poisson brackets

DG Poisson algebra and its universal enveloping algebra

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