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In this thesis we shall discuss some properties of centralisers in classical Lie algebas and related structures. Let K be an algebraically closed field of characteristic p greater than or equal to 0. Let G be a simple algebraic group over K. We shall denote by g = Lie(G) the Lie algebra of G, and for x in g denote by g_x the centraliser. Our results follow three distinct but related themes: the modular representation theory of centralisers, the sheets of simple Lie algebras and the representation theory of finite W-algebras and enveloping algebras. When G is of type A or C and p > 0 is a good prime for G, we show that the invariant algebras S(g_x)^{G_x} and U(g_x)^{G_x} and polynomial algebras on rank g generators, that the algebra S(g_x)^{g_x} is generated by S(g_x)^p and S(g_x)^{G_x}, whilst U(g_x)^{g_x} is generated by U(g_x)^{G_x} and the p-centre, generalising a classical theorem of Veldkamp. We apply the latter result to confirm the first Kac-Weisfeiler conjecture for g_x, giving a precise upper bound for the dimensions of simple U(g_x)-modules. This allows us to characterise the smooth locus of the Zassenhaus variety in algebraic terms. These results correspond to an article, soon to appear in the Journal of Algebra. The results of the next chapter are particular to the case x nilpotent with G connected of type B, C or D in any characteristic good for G. Our discussion is motivated by the theory of finite W-algebras which shall occupy our discussion in the final chapter, although we make several deductions of independent interest. We begin by describing a vector space decomposition for [g_x g_x] which in turn allows us to give a formula for dim g_x^\ab where g_x^\ab := g_x / [g_x g_x]. We then concoct a combinatorial parameterisation of the sheets of g containing x and use it to classify the nilpotent orbits lying in a unique sheet. We call these orbits non-singular. Subsequently we give a formula for the maximal rank of sheets containing x and show that it coincides with dim g_x^\ab if and only if x is non-singular. The latter result is applied to show for any (not necessarily nilpotent) x in g lying in a unique sheet, that the orthogonal complement to [g_x g_x] is the tangent space to the sheet, confirming a recent conjecture. In the final chapter we set p = 0 and consider the finite W-algebra U(g,x), again with G of type B, C or D. The one dimensional representations are parameterised by the maximal spectrum of the maximal abelian quotient E = Specm U(g, x)^\ab and we classify the nilpotent elements in classical types for which E is isomorphic to an affine space A^d_K: they are precisely the non-singular elements of the previous chapter. The component group acts naturally on E and the fixed point space lies in bijective correspondence with the set of primitive ideals of U(g) for which the multiplicity of the correspoding primitive quotient is one. We call them multiplicity free. We show that this fixed point space is always an affine space, and calculate its dimension. Finally we exploit Skryabin's equivalence to study parabolic induction of multiplicity free ideals. In particular we show that every multiplicity free ideals whose associated variety is the closure of an induced orbit is itself induced from a completely prime primitive ideals with nice properties, generalising a theorem of Moeglin. The results of the final two chapters make up a part of a joint work with Alexander Premet.

In the present thesis we study free centre-by-(abelian-by-exponent 2) groups. These are in the class of free centre-by-metabelian groups which in turn are a special case of quotients of the form F=[R0; F] where F is a free group, R is a normal subgroup of F and R0 is the commutator subgroup of R. The latter have been an object of investigation for more than forty years, due to their intriguing feature of having non-trivial torsion under certain conditions. This was first discovered for the case where R = F0. For arbitrary F and R, if there is torsion in F=[R0; F], it is bound to be contained in the central subgroup R0=[R0; F] which decomposes into a direct sum of a free abelian group and a (possibly trivial) torsion group of exponent dividing 4. If F=R has no elements of order 2, then the torsion subgroup is isomorphic to the homology group H4(F=R;Z2). Thus the question that remains open is what happens if F=R contains elements of order 2.