dc.contributor.author |
Awala, Hussein Abdallah. |
dc.date.accessioned |
2012-12-03T13:33:30Z |
dc.date.available |
2012-12-03T13:33:30Z |
dc.date.issued |
2012 |
dc.identifier.uri |
http://hdl.handle.net/10938/9348 |
dc.description |
Thesis (M.S.)--American University of Beirut, Department of Mathematics, 2012.;"Advisor : Dr. Nahlus, Nazih, Professor, Mathematics--Members of Committee : Dr. Abu Khuzam Hazar, Professor, Mathematics Dr. Azar, Monique, Assistant Professor, Mathematics." |
dc.description |
Includes bibliographical references (leaf 76) |
dc.description.abstract |
We investigate some applications of ultraproducts in Algebra. In particular, we first present the classical applications: Robinson Theorem and Malcev Theorem. Then we focus on the most recent application of ultraproducts by G. Bergman and N. Nahlus. For example, we show that any finite-dimensional quotient of an infinite direct product (over any arbitrary index set) of finite-dimensional solvable Lie algebras is also solvable. The same is true for nilpotent and semi-simple Lie algebras. However, the proof in the case of semisimple Lie algebras, requires the deep theorem that L=[x,L] +[y, L] for some x, y in L , or it requires Brown Theorem which we both prove. The general technique in all three cases requires an investigation of ultraproducts with (resp. without) countably-complete ultrafilters. For simplicity, we shall assume that the base field is algebraically closed of characteristic 0. |
dc.format.extent |
vii, 76 leaves 30 cm. |
dc.relation.ispartof |
Theses, Dissertations, and Projects |
dc.subject.classification |
T:005653 AUBNO |
dc.subject.lcsh |
Lie algebras.;Ultraproducts. |
dc.title |
Ultraproducts and quotients of infinite direct products of lie algebras / by Hussein Abdallah Awala. |
dc.type |
Thesis |
dc.contributor.department |
American University of Beirut. Faculty of Arts and Sciences. Department of Mathematics. |