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

Abstract (Summary)

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.
Bibliographical Information:

Advisor:Tom Lada; Larry Norris; Steve Schecter; Ron Fulp

School:North Carolina State University

School Location:USA - North Carolina

Source Type:Master's Thesis

Keywords:mathematics

ISBN:

Date of Publication:11/13/2003