The study of deformations of an algebra has been a topic of interest for quite some time, since it allows us to not only produce new algebras but also better understand the original algebra. Given an algebra, finding all its deformations is, if at all possible, quite a challenging problem. For this reason, several specializations of this question have been proposed. For instance, some authors concentrate their efforts in the study of deformations of an algebra arising from an action of a Hopf algebra. The purpose of this dissertation is to discuss a general construction of a deformation of a smash product algebra coming from an action of a particular Hopf algebra. This Hopf algebra is generated by skew-primitive and group-like elements, and depends on a complex parameter. The smash product algebra is defined on the quantum symmetric algebra of a nite-dimensional vector space and a group. In particular, an application of this result has enabled us to find a deformation of such a smash product algebra which is, to the best of our knowledge, the first known example of a deformation in which the new relations in the deformed algebra involve elements of the original vector space. Finally, using Hochschild cohomology, we show that these deformations are nontrivial.

In this work we investigate the structure of this type of Lie algebras over arbitrary fields F by constructing them from their Abelian ideal. To accomplish this, an algorithm is developed and as application a classification up to 7-dimensional Lie Algebras is given. We discuss a recent example of financial mathematics as well.

Thesis (Ph. D.)--University of Oregon, 2001. / Typescript. Includes vita and abstract. Includes bibliographical references (leaves 62-63). Also available for download via the World Wide Web; free to University of Oregon users.

Thesis (Ph. D.)--University of Oregon, 2003. / Typescript. Includes vita and abstract. Includes bibliographical references (leaves 106-108). Also available for download via the World Wide Web; free to University of Oregon users.

Thesis (M. Phil.)--University of Hong Kong, 2001. / Includes bibliographical references (leaves 14-15).

Thesis (Ph. D.)--University of Oregon, 2002. / Typescript. Includes vita and abstract. Includes bibliographical references (leaf 77). Also available for download via the World Wide Web; free to University of Oregon users.

Much work has been done on the ℓ¹-algebras of groups, but much less on ℓ¹-algebras of semigroups. This thesis studies those of inverse semigroups, also known as generalised groups, with emphasis on the involutive structure. Where results extend to the semigroup ring, I extend them. I determine the characters of a semilattice in terms of its order structure. The simplest suffice to separate its ℓ¹-algebra. I also determine the algebra's minimal idempotents. I introduce a generalisation of Banach *-algebras which has good hereditary properties and includes the inverse semi groups rings. These latter have an ultimate identity which can be used to test for representability. Involutive semigroups with s*s an idempotent yield inverse semi groups when quotiented by the congruence induced by their algebras' *-radical. The left regular *-representation of inverse seroigroups is faithful and acts like that of groups. The corresponding idea of amenability coincides with the traditional one. Brandt semi groups have the weak containment property iff the associated group does. The relationship of ideals to weak containment is studied, and inverse semigroups with well ordered semilattices are shown to have the property if all their subgroups do. The converse is extended for Clifford semigroups. Symmetry and related ideas are considered, and basic results proved for the above mentioned generalisation, and a better version for a possibly more restricted generalisation. The symmetry of an ℓ¹-algebra of an E-unitary inverse semi group is shown to depend on the symmetry of the ℓ¹-algebra of its maximal group homomorphic image if the semilattice has a certain structure or the semigroup is a Clifford semigroup. Inverse semi groups with well ordered semilattices are shown to have symmetric ℓ¹-algebra if all the subgroups do. Finally, some topologically simple ℓ¹-algebras and simple semigroup rings are constructed, extending results on simple inverse semigroup rings.

published_or_final_version / Mathematics / Doctoral / Doctor of Philosophy

Not available / text

No description available.