JournalsjemsVol. 16, No. 2pp. 387–407

One-parameter contractions of Lie-Poisson brackets

  • Oksana Yakimova

    Friedrich-Schiller-Universität Jena, Germany
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Abstract

We consider contractions of Lie and Poisson algebras and the behaviour of their centres under contractions. A polynomial Poisson algebra A=K[An]{\mathcal A}=\mathbb K[\mathbb A^n] is said to be of Kostant type, if its centre Z(A)Z({\mathcal A}) is freely generated by homogeneous polynomials F1,,FrF_1,\ldots,F_r such that they give Kostant's regularity criterion on An\mathbb A^n (dxFid_xF_i are linear independent if and only if the Poisson tensor has the maximal rank at xx). If the initial Poisson algebra is of Kostant type and FiF_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\gt g corresponding to a decomposition >g=>hV\gt g=\gt h \oplus V, where >h\gt h is a subalgebra. Here A=S(>g)=K[>g]{\mathcal A}={\mathcal S}(\gt g)=\mathbb K[\gt g^*], Z(A)=S(>g)>gZ({\mathcal A})={\mathcal S}(\gt g)^\gt g, and the contracted Lie algebra is a semidirect product of >h\gt h and an Abelian ideal isomorphic to >g/>h\gt g/\gt h as an >h\gt h-module. In the first example, >h\gt h is a symmetric subalgebra and in the second, it is a Borel subalgebra and VV is the nilpotent radical of an opposite Borel.

Cite this article

Oksana Yakimova, One-parameter contractions of Lie-Poisson brackets. J. Eur. Math. Soc. 16 (2014), no. 2, pp. 387–407

DOI 10.4171/JEMS/436