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Let $g$ be a classical Lie algebra, i.e., either $gl_n$, $sp_n$, or $so_n$and let $e\in g$ be a nilpotent element. We study various properties ofcentralisers $g_e$. The first four sections deal with rather elementaryquestions, like the centre of $g_e$, commuting varieties associated with $g_e$,or centralisers of commuting pairs. The second half of the paper addressesproblems related to different Poisson structures on $g_e^*$ and symmetricinvariants of $g_e$.

Surprising properties of centralisers in classical Lie algebras