RECORD
Poisson and symplectic functions in Lie algebroid theory
Symplectic Geometry · Submitted 13 Nov 2007
abstract compiled · 2026-09-23 · no language-model review
The authors' main claim Emphasizing the role of Gerstenhaber algebras and of higher derived brackets in the theory of Lie algebroids, we show that the several Lie algebroid brackets which have been introduced in the recent literature can all be defined in terms of Poisson and pre-symplectic functions in the sense of Roytenberg and Terashima.
The compiled record
The paper's own sentences, arranged. Each label is drawn from a closed vocabulary; the number beside it is the probability the classifier gave that label. Numbers in the tables are the authors', checked by code. No sentence in this record was written by a model.
Open the compiled record
The abstract, sentence by sentence
Emphasizing the role of Gerstenhaber algebras and of higher derived brackets in the theory of Lie algebroids, we show that the several Lie algebroid brackets which have been introduced in the recent literature can all be defined in terms of Poisson and pre-symplectic functions in the sense of Roytenberg and Terashima. We prove that in this very general framework there exists a one-to-one correspondence between non-degenerate Poisson functions and symplectic functions. We determine the differential associated to a Lie algebroid structure obtained by twisting a structure with background by both a Lie bialgebra action and a Poisson bivector.
background contribution method result interpretation main claim
| # | sentence | role | the authors' own work | states the main conclusion | strength |
|---|---|---|---|---|---|
| s0 | Emphasizing the role of Gerstenhaber algebras and of higher derived brackets in the theory of Lie algebroids, we show that the several Lie algebroid brackets which have been introduced in the recent literature can all be defined in terms of Poisson and pre-symplectic functions in the sense of Roytenberg and Terashima. | contribution 0.88 | 0.90 | 0.68 | asserted 1.00 |
| s1 | We prove that in this very general framework there exists a one-to-one correspondence between non-degenerate Poisson functions and symplectic functions. | result 1.00 | 0.95 | 0.34 | asserted 1.00 |
| s2 | We determine the differential associated to a Lie algebroid structure obtained by twisting a structure with background by both a Lie bialgebra action and a Poisson bivector. | result 0.80 | 0.94 | 0.10 | asserted 1.00 |
What was examined, and what was not
- arXiv listing abstract v4
- full text
- figures and tables
- references and citing sentences
- relations to other papers
- P abstract
- S sections
- T body sentences
- C citations
- R cross-paper relations
The paper
Original abstract
Emphasizing the role of Gerstenhaber algebras and of higher derived brackets in the theory of Lie algebroids, we show that the several Lie algebroid brackets which have been introduced in the recent literature can all be defined in terms of Poisson and pre-symplectic functions in the sense of Roytenberg and Terashima. We prove that in this very general framework there exists a one-to-one correspondence between non-degenerate Poisson functions and symplectic functions. We determine the differential associated to a Lie algebroid structure obtained by twisting a structure with background by both a Lie bialgebra action and a Poisson bivector.
Human review
References
How the literature uses it
9 arXiv papers cite this in 18 sentences
Papers citing it · 13 sentences without a confident label
Builds on it 1 paper
-
The following proposition [ref] [ref] [ref] extends results of [ref] .
Dirac pairs · Poisson and presymplectic functions on proto-bialgebroids · 0.99
Uses its method or tool 2 papers
-
Our description of Nijenhuis structures and related concepts relies on the use of Roytenberg's graded Poisson bracket on the minimal symplectic realization of a Courant algebroid [ref] , and on its interplay with the big bracket [ref] [ref] [ref] [ref] when the Courant algebroid is the double of a Lie bialgebroid, and, in particular, in the case of the generalized tangent bundle of a manifold.
Nijenhuis structures on Courant algebroids · Introduction · 0.99
-
We use twisting of a homological function introduced by [ref] and analyzed in [ref] .
Compatible $E$-Differential Forms on Lie Algebroids over (Pre-)Multisymplectic Manifolds · Twisting of homological functions and twisted QP-manifolds · 0.96
Cites it as background 2 papers
-
This special kind of canonical function of degree $2$ is called a Poisson function in [ref] [ref] .
Canonical Functions and Differential Graded Symplectic Pairs in Supergeometry and AKSZ Sigma Models with Boundary · Canonical Functions on QP Manifolds · 0.91
-
The minimal symplectic realization of $E$ is $E = T^*[2]A[1]$, and the canonical Poisson bracket of $$ coincides with the big bracket (for which see [ref] [ref] [ref] ), which we also denote by ${ , }$.
Nijenhuis structures on Courant algebroids · The double of a Lie bialgebroid · 0.93
What it cites
Cites as background 1 paper
-
The particular case of the quasi-Poisson $𝔤$-spaces was recently treated in [ref] .
Dirac geometry, quasi-Poisson actions and D/G-valued moment maps · The twisted differential · 0.90
Discussion
Know something about this paper? Leave a comment, or highlight any passage above to comment on it.