The objective of this dissertation is to give a description of the non-degenerate invariant bilinear forms on finite-dimensional Lie algebras over fields of characteristic zero which do not generally arise as trace forms Chapter I provides the necessary background information on Lie algebras Chapter II introduces the notion of invariant bilinear forms, with special attention given to trace forms Chapter III is devoted to a general presentation of L-modules with a non-degenerate invariant bilinear form, particularly in relation to duality in the presence of a non-degenerate bilinear form. The first part of the chapter exploits the annihilator ideal in greater detail Chapter IV treats invariant bilinear forms on simple and semi-simple Lie algebras. For simple Lie algebras L, all non-zero invariant bilinear forms are non-degenerate and, if F is an algebraically closed field, every such form is an F-multiple of the Killing form (kappa) of L. It is shown that this result still holds if F is replaced by the field of real numbers Chapter V provides background information on nilpotent and solvable Lie algebras. Then it is shown that a solvable Lie algebra can carry a non-degenerate invariant bilinear form only if it is of a special type. The theory of ground ring extensions developed here will lead to a construction of nilpotent Lie algebras of arbitrary nilpotency class with a non-degenerate invariant bilinear form Chapter VI analyzes the structure of finite-dimension Lie algebras L over fields of characteristic zero with a non-degenerate invariant bilinear form. The results of this investigation lead to a description of the non-degenerate invariant bilinear forms on L / acase@tulane.edu
| Identifer | oai:union.ndltd.org:TULANE/oai:http://digitallibrary.tulane.edu/:tulane_26977 |
| Date | January 1984 |
| Contributors | Keith, Verena Sabine (Author) |
| Publisher | Tulane University |
| Source Sets | Tulane University |
| Language | English |
| Detected Language | English |
| Rights | Access requires a license to the Dissertations and Theses (ProQuest) database., Copyright is in accordance with U.S. Copyright law |
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