Title page for ETD etd-11092003-154027

Type of Document

Dissertation

Author

Al-Ashhab, Samer Shafiq,

URN

etd-11092003-154027

Title

THE ROLE OF SH-LIE ALGEBRAS IN LAGRANGIAN FIELD THEORY.

Degree

PhD

Graduate Program

Mathematics

Advisory Committee

Advisor Name	Title
Ron Fulp	Committee Chair
Larry Norris	Committee Member
Steve Schecter	Committee Member
Tom Lada	Committee Member

Keywords

Lie algebra
sh-Lie algebra
reduction
Poisson bracket
group action
jet bundle
manifold

Date of Defense

2003-10-30

Availability

unrestricted

Abstract

The purpose of this dissertation is to study strongly homotopy Lie algebras
(sh-Lie algebras) and their applications with primary emphasis on applications
to field theory. Strongly homotopy Lie algebras are defined on graded vector
spaces. They generally consist of an infinite sequence of mappings
$l_1,l_2,l_3,cdots$, which satisfy certain identities. We show that, in the
presence of appropriate hypotheses, there always exists
a simplified sh-Lie algebra structure with $l_n=0$ for $n>3$.
This is a special case which has occured in several applications. While it is
known that sh-Lie algebras arise in field theory as a homological resolution of
a Poisson bracket defined on the space of local functionals, we show how these
sh-Lie algebras transform in the event of canonical transformations on the
space of local functionals. Additionally, it is shown how a group which acts
via canonical transformations transforms the sh-Lie structure and eventually
leads to reduction theorems. Two kinds of reduction are obtained corresponding
to two different kinds of group action and, in each case it is shown how to
obtain an induced sh-Lie algebra on a corresponding reduced graded vector
space. Several applications of the theory are considered as well.

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