# One-parameter contractions of Lie-Poisson brackets

Journal of the European Mathematical Society (2014)

- Volume: 016, Issue: 2, page 387-407
- ISSN: 1435-9855

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topYakimova, 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 -

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