The Role of sh-Lie Algebras in Lagrangian Field Theory.

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.