Advanced Search

Let $𝔤$ be a classical Lie algebra, , either ${\mathrm{𝔤𝔩}}_n$, ${\mathrm{𝔰𝔭}}_n$, or ${\mathrm{𝔰𝔬}}_n$ and let $e$ be a nilpotent element of $𝔤$. We study various properties of the centralisers ${𝔤}_e$. The first four sections deal with rather elementary questions, like the centre of ${𝔤}_e$, commuting varieties associated with ${𝔤}_e$, or centralisers of commuting pairs. The second half of the paper addresses problems related to different Poisson structures on ${𝔤}_e^{*}$ and symmetric invariants of ${𝔤}_e$.

We consider contractions of Lie and Poisson algebras and the behaviour of their centres under contractions. A polynomial Poisson algebra $𝒜=𝕂\left[{𝔸}^n\right]$ is said to be of Kostant type, if its centre $Z\left(𝒜\right)$ is freely generated by homogeneous polynomials $F_1,...,F_r$ such that they give Kostant’s regularity criterion on ${𝔸}^n(d_x{F}_i$ 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...

Recently, E.Feigin introduced a very interesting contraction $𝔮$ of a semisimple Lie algebra $𝔤$ (see arXiv:1007.0646 and arXiv:1101.1898). We prove that these non-reductive Lie algebras retain good invariant-theoretic properties of $𝔤$. For instance, the algebras of invariants of both adjoint and coadjoint representations of $𝔮$ are free, and also the enveloping algebra of $𝔮$ is a free module over its centre.