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In this thesis it is shown that the isodiametric inequality fails for Carnot-Caratheodory balls in the Heisenberg group W (n E N). Estimates for the ratio of the volume of a ball to the maximal volume of a set of the same diameter are established in this group, and the set of the maximal volume is also found among all sets of revolution about the vertical axis having the same diameter. Results of the similar nature are obtained in the additive group Rn+1 (n G N) with non-isotropic dilations. Using a connection between the isodiametric problem and the Besicovitch 1/2-problem it is proved that the generalized Besicovitch 1/2-conjecture fails in the Heisenberg group HP (1 < n < 8) of the Hausdorff dimension 2n+2 and the additive group Rn+1 (n £ N) having non-isotropic dilations and integer Hausdorff dimension greater than or equal to n + 2. But the 1-dimensional case is shown to be exceptional - the generalized Besicovitch 1/2-conjecture is true in any locally compact group which is equipped with an invariant metric, its Haar measure and has the Hausdorff dimension 1. A question about the relation among the Hausdorff, the spherical and the centred Hausdorff measures of codimension one restricted to a smooth surface is also investigated in the Heisenberg group M1. It is proved that these measures differ but coincide up to positive constant multiples, estimates for which are found.

Given a collection of matrix similarity classes Cl, ... , Ck the additive matrix problem asks under what conditions do there exist matrices Ai E Cj for j = 1, ... , k such that Al + ' .. + Ak = O. This and similar problems have been examined under various guises in the literature. The results of Crawley-Boevey use the representation theory of quivers to link the additive matrix problem to the root systems of quivers. We relate the results of Crawley-Boevey to another partial solution offered by Silva et al. and develop some tools to interpret the solutions of Silva et al. in terms of root systems. The results of Crawley-Boevey require us to know the precise Jordan form of the similarity classes; we address the problem of invoking Crawley-Boevey's results when only the invariant polynomials are known and we are not permitted to use polynomial factorization. We use the machinery of symmetric quivers and symmetric representations to study the problem of finding symmetric matrix solutions to the additive matrix problem. We show the reflection functors, defined for representations of deformed preprojective algebras, can be defined for symmetric representations. We show every rigid irreducible solution to the additive matrix problem can be realized by symmetric matrices and we use algebraic geometry to show that in some circumstances there are solutions which cannot be realized by symmetric matrices. We show there exist symmetric representations of deformed preprojective algebras of foot dimension vectors when the underlying quiver is Dynkin or extended Dynkin of type An Of Dn.

In recent years, two generalisations of the theory of Lie algebras have become prominent, namely the "semi-classical" theory of Lie bialgebras and the "quantum" theory of Bopf algebras, including the quantized enveloping algebras. I develop an inductive approach to the study of these objects. An important tool is a construction called double-bosonisation defined by Majid for both Lie bialgebras and Hopf algebras, inspired by the triangular decomposition of a Lie algebra into positive and negative roots and a Cartan subalgebra. We describe two specific applications. The first uses double-bosonisation to add positive and negative roots and considers the relationship between two algebras when there is an inclusion of the associated Dynkin diagrams. In this setting, which we call Lie induction, doublebosonisation realises the addition of nodes to Dynkin diagrams. We use our methods to obtain necessary conditions for such an induction to be simple, using representation theory, providing a different perspective on the classification of simple Lie algebras. We consider the corresponding scheme for quantized enveloping algebras, based on inclusions of the associated root data. We call this quantum Lie induction. We prove that we have a double-bosonisation associated to these inclusions and investigate the structure of the resulting objects, which are Hopf algebras in braided categories, that is, covariant Bopf algebras. The second application generalises one of the most important constructions in this field, namely the Drinfel'd double of a Lie bialgebra, which has dimension twice that of the underlying algebra. Our construction, the triple, has dimension three times that of the input algebra. Our main result is that when the input algebra is factorisable, this is isomorphic to the triple direct sum as an algebra and a twisting as a coalgebra. We also indicate a number of ways in which the triple is related to the double.