Canonical Poisson-Nijenhuis structures on higher order tangent bundles

P. M. Kouotchop Wamba. "Canonical Poisson-Nijenhuis structures on higher order tangent bundles." Annales Polonici Mathematici 111.1 (2014): 21-37. <http://eudml.org/doc/280896>.

@article{P2014,
abstract = {Let M be a smooth manifold of dimension m>0, and denote by $S_\{can\}$ the canonical Nijenhuis tensor on TM. Let Π be a Poisson bivector on M and $Π^\{T\}$ the complete lift of Π on TM. In a previous paper, we have shown that $(TM, Π^\{T\}, S_\{can\})$ is a Poisson-Nijenhuis manifold. Recently, the higher order tangent lifts of Poisson manifolds from M to $T^\{r\}M$ have been studied and some properties were given. Furthermore, the canonical Nijenhuis tensors on $T^\{A\}M$ are described by A. Cabras and I. Kolář [Arch. Math. (Brno) 38 (2002), 243-257], where A is a Weil algebra. In the particular case where $A= J^\{r\}₀(ℝ, ℝ) ≃ ℝ^\{r+1\}$ with the canonical basis $(e_\{α\})$, we obtain for each 0 ≤ α ≤ r the canonical Nijenhuis tensor $S_\{α\}$ on $T^\{r\}M$ defined by the vector $e_\{α\}$. The tensor $S_\{α\}$ is called the canonical Nijenhuis tensor on $T^\{r\}M$ of degree α. In this paper, we show that if (M,Π) is a Poisson manifold, then for each α with 1 ≤ α ≤ r, $(T^\{r\}M, Π^\{(c)\}, S_\{α\})$ is a Poisson-Nijenhuis manifold. In particular, we describe other prolongations of Poisson manifolds from M to $T^\{r\}M$ and we give some of their properties.},
author = {P. M. Kouotchop Wamba},
journal = {Annales Polonici Mathematici},
keywords = {Nijenhuis manifold; vertical lift of vector fields; complete lift of Lie algebroids},
language = {eng},
number = {1},
pages = {21-37},
title = {Canonical Poisson-Nijenhuis structures on higher order tangent bundles},
url = {http://eudml.org/doc/280896},
volume = {111},
year = {2014},
}

TY - JOUR
AU - P. M. Kouotchop Wamba
TI - Canonical Poisson-Nijenhuis structures on higher order tangent bundles
JO - Annales Polonici Mathematici
PY - 2014
VL - 111
IS - 1
SP - 21
EP - 37
AB - Let M be a smooth manifold of dimension m>0, and denote by $S_{can}$ the canonical Nijenhuis tensor on TM. Let Π be a Poisson bivector on M and $Π^{T}$ the complete lift of Π on TM. In a previous paper, we have shown that $(TM, Π^{T}, S_{can})$ is a Poisson-Nijenhuis manifold. Recently, the higher order tangent lifts of Poisson manifolds from M to $T^{r}M$ have been studied and some properties were given. Furthermore, the canonical Nijenhuis tensors on $T^{A}M$ are described by A. Cabras and I. Kolář [Arch. Math. (Brno) 38 (2002), 243-257], where A is a Weil algebra. In the particular case where $A= J^{r}₀(ℝ, ℝ) ≃ ℝ^{r+1}$ with the canonical basis $(e_{α})$, we obtain for each 0 ≤ α ≤ r the canonical Nijenhuis tensor $S_{α}$ on $T^{r}M$ defined by the vector $e_{α}$. The tensor $S_{α}$ is called the canonical Nijenhuis tensor on $T^{r}M$ of degree α. In this paper, we show that if (M,Π) is a Poisson manifold, then for each α with 1 ≤ α ≤ r, $(T^{r}M, Π^{(c)}, S_{α})$ is a Poisson-Nijenhuis manifold. In particular, we describe other prolongations of Poisson manifolds from M to $T^{r}M$ and we give some of their properties.
LA - eng
KW - Nijenhuis manifold; vertical lift of vector fields; complete lift of Lie algebroids
UR - http://eudml.org/doc/280896
ER -