One-parameter contractions of Lie-Poisson brackets

Yakimova, Oksana. "One-parameter contractions of Lie-Poisson brackets." Journal of the European Mathematical Society 016.2 (2014): 387-407. <http://eudml.org/doc/277396>.

@article{Yakimova2014,
abstract = {We consider contractions of Lie and Poisson algebras and the behaviour of their centres under contractions. A polynomial Poisson algebra $\{\mathcal \{A\}\}=\mathbb \{K\}[\mathbb \{A\}^n]$ is said to be of Kostant type, if its centre $Z(\{\mathcal \{A\}\})$ is freely generated by homogeneous polynomials $F_1,\ldots ,F_r$ such that they give Kostant’s regularity criterion on $\mathbb \{A\}^n (d_xF_i$ are linear independent if and only if the Poisson tensor has the maximal rank at $x$). If the initial Poisson algebra is of Kostant type and $F_i$ satisfy a certain degree-equality, then the contraction is also of Kostant type. The general result is illustrated by two examples. Both are contractions of a simple Lie algebra $> g$ corresponding to a decomposition $> g=> h \oplus V$, where $>h$ is a subalgebra. Here $\{\mathcal \{A\}\}=\{\mathcal \{S\}\}(> g)=\mathbb \{K\}[> g^*], Z(\{\mathcal \{A\}\})=\{\mathcal \{S\}\}(> g)^\{> g\}$, and the contracted Lie algebra is a semidirect product of $>h$ and an Abelian ideal isomorphic to $>g/>h$ as an $>h$-module. In the first example, $>h$ is a symmetric subalgebra and in the second, it is a Borel subalgebra and $V$ is the nilpotent radical of an opposite Borel.},
author = {Yakimova, Oksana},
journal = {Journal of the European Mathematical Society},
keywords = {Nilpotent orbits; centralisers; symmetric invariants; nilpotent orbits; centralisers; symmetric invariants},
language = {eng},
number = {2},
pages = {387-407},
publisher = {European Mathematical Society Publishing House},
title = {One-parameter contractions of Lie-Poisson brackets},
url = {http://eudml.org/doc/277396},
volume = {016},
year = {2014},
}

TY - JOUR
AU - Yakimova, Oksana
TI - One-parameter contractions of Lie-Poisson brackets
JO - Journal of the European Mathematical Society
PY - 2014
PB - European Mathematical Society Publishing House
VL - 016
IS - 2
SP - 387
EP - 407
AB - We consider contractions of Lie and Poisson algebras and the behaviour of their centres under contractions. A polynomial Poisson algebra ${\mathcal {A}}=\mathbb {K}[\mathbb {A}^n]$ is said to be of Kostant type, if its centre $Z({\mathcal {A}})$ is freely generated by homogeneous polynomials $F_1,\ldots ,F_r$ such that they give Kostant’s regularity criterion on $\mathbb {A}^n (d_xF_i$ are linear independent if and only if the Poisson tensor has the maximal rank at $x$). If the initial Poisson algebra is of Kostant type and $F_i$ satisfy a certain degree-equality, then the contraction is also of Kostant type. The general result is illustrated by two examples. Both are contractions of a simple Lie algebra $> g$ corresponding to a decomposition $> g=> h \oplus V$, where $>h$ is a subalgebra. Here ${\mathcal {A}}={\mathcal {S}}(> g)=\mathbb {K}[> g^*], Z({\mathcal {A}})={\mathcal {S}}(> g)^{> g}$, and the contracted Lie algebra is a semidirect product of $>h$ and an Abelian ideal isomorphic to $>g/>h$ as an $>h$-module. In the first example, $>h$ is a symmetric subalgebra and in the second, it is a Borel subalgebra and $V$ is the nilpotent radical of an opposite Borel.
LA - eng
KW - Nilpotent orbits; centralisers; symmetric invariants; nilpotent orbits; centralisers; symmetric invariants
UR - http://eudml.org/doc/277396
ER -