In this thesis we investigate the structure of locally nilpotent 5-Engel p-groups. We show that for p > 7, locally nilpotent 5-Engel p-groups have class at most 10. This is a global theorem, where the result is not dependent on the number of generators of the group. The proof uses new and established Lie methods and a custom C++ implementation of an algorithm that constructs minimal generating sets and structure constants of multi- graded Lie algebras in a variety defined by three multilinear relations, which hold in the Lie rings associated with 5-Engel p-groups. We obtain our results by calculating in the set Q(p) = {~ I x E Z, yE Z+, Y # 0 modulo any p f/. p} (where p is a set of excluded primes and x, y are arbitrarily large integers), as well as the fields Zp, p prime. We introduce several reduction theorems, making the result possible. We also present results about the normal closure of elements in these groups. We use a Higman reduction theorem and the same custom C++ program to show that locally nilpotent 5-Engel p-groups, p 2: 5, are Fitting, with Fitting degree at most 4 if p > 7, at most 5 if p = 7 and at most 6 if p = 5. These results are best possible.

Thesis. 1976. Ph.D.--Massachusetts Institute of Technology. Dept. of Mathematics. / Microfiche copy available in Archives and Science. / Vita. / Bibliography: leaves 184-186. / by David Vogan. / Ph.D.

So, Tse Leung. / Thesis (M.Phil.)--Chinese University of Hong Kong, 2010. / Includes bibliographical references (leaves 55-56). / Abstracts in English and Chinese. / Chapter 1 --- Geometric Quantization --- p.8 / Chapter 1.1 --- Motivation --- p.8 / Chapter 1.2 --- Prequantization --- p.10 / Chapter 1.3 --- Kahler Polarization --- p.14 / Chapter 1.4 --- Holomorphic Quantization and Fock Space --- p.17 / Chapter 1.4.1 --- Example: Fock Space --- p.19 / Chapter 2 --- Projectively Flat Connection --- p.21 / Chapter 2.1 --- Variation of Complex Structure --- p.21 / Chapter 2.2 --- Projectively Flat Connection on H --- p.24 / Chapter 2.2.1 --- Example of n= 1: --- p.30 / Chapter 3 --- Construction of Projectively Flat Connection --- p.34 / Chapter 3.1 --- Mechanism for Construction of Projectively Flat Con- nection --- p.34 / Chapter 3.2 --- K = C Case --- p.39 / Chapter 3.2.1 --- Identification of C´ؤlinear Complex Structures --- p.40 / Chapter 3.2.2 --- Projecitvely flat bundle on U(n)×U(n) --- p.43 / Chapter 3.2.3 --- Example of n=1 --- p.44 / Chapter 3.3 --- K = H Case --- p.45 / Chapter 3.3.1 --- Identification of H-linear Complex Structures --- p.46 / Chapter 3.3.2 --- Projecitvely flat bundle on SO*(4n) --- p.49 / Chapter 3.3.3 --- Example of n = 1 --- p.51 / Bibliography --- p.55

An important object of study in harmonic analysis is the heat equation. On a Euclidean space, the fundamental solution of the associated semigroup is known as the heat kernel, which is also the law of Brownian motion. Similar statements also hold in the case of a Lie group. By using the wrapping map of Dooley and ildberger, we show how to wrap a Brownian motion to a compact Lie group from its Lie algebra (viewed as a Euclidean space) and find the heat kernel. This is achieved by considering It??o type stochastic differential equations and applying the Feynman-Ka??c theorem. We also consider wrapping Brownian motion to various symmetric spaces, where a global generalisation of Rouvi`ere???s formula and the e-function are considered. Additionally, we extend some of our results to complex Lie groups, and certain non-compact symmetric spaces.

Représentations des groupes de Lie conformes et quantification des espaces symétriques

This thesis is an expository account of three central theorems in the representation theory of semisimple Lie groups, namely the theorems of Borel-Weil-Bott, Casselman-Osborne and Kostant. The first of these realizes all the irreducible holomorphic representations of a complex semisimple Lie group G in the cohomology of certain sheaves of equivariant line bundles over the flag variety of G. The latter two theorems describe the Lie algebra cohomology of a maximal nilpotent subalgebra of Lie(G) with coefficients in an irreducible Lie(G)-module. Applications to geometry and representation theory are given. Also included is a brief overview of Schmid's far-reaching generalization of the Borel--Weil--Bott theorem to the setting of unitary representations of real semisimple Lie groups on (possibly infinite-dimensional) Hilbert spaces.

Thesis (Ph. D.)--University of Hong Kong, 2000. / Includes bibliographical references (leaves 50-53).

Thesis (M. Phil.)--University of Hong Kong, 2010. / Includes bibliographical references (leaves 68-70). Also available in print.

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