A general construction of an sh Lie algebra (L ∞ -algebra) from a homological resolution of a Lie algebra is given. It is applied to the space of local functionals equipped with a Poisson bracket, induced by a bracket for local functions along the lines suggested by Gel'fand, Dickey and Dorfman. In this way, higher order maps are constructed which combine to form an sh Lie algebra on the graded differential algebra of horizontal forms. The same construction applies for graded brackets in field theory such as the Batalin-Fradkin-Vilkovisky bracket of the Hamiltonian BRST theory or the Batalin-Vilkovisky antibracket.

Traditionally symmetries of field theories are encoded via Lie group actions, or more generally, as Lie algebra actions. A significant generalization is required when 'gauge parameters' act in a field dependent way. Such symmetries appear in several field theories, most notably in a 'Poisson induced' class due to Schaller and Strobl [SS94] and to Ikeda[Ike94], and employed by Cattaneo and Felder [CF99] to implement Kontsevich's deformation quantization [Kon97]. Consideration of 'particles of spin > 2 led Berends, Burgers and van Dam [Bur85, BBvD84, BBvD85] to study 'field dependent parameters' in a setting permitting an analysis in terms of smooth functions. Having recognized the resulting structure as that of an sh-lie algebra (L ∞ -algebra), we have now formulated such structures entirely algebraically and applied it to a more general class of theories with field dependent symmetries.

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In many Lagrangian field theories, there is a Poisson bracket on the space of local functionals. One may identify the fields of such theories as sections of a vector bundle. It is known that the Poisson bracket induces an sh-Lie structure on the graded space of horizontal forms on the jet bundle of the relevant vector bundle. We consider those automorphisms of the vector bundle which induce mappings on the space of functionals preserving the Poisson bracket and refer to such automorphisms as canonical automorphisms.We determine how such automorphisms relate to the corresponding sh-Lie structure. If a Lie group acts on the bundle via canonical automorphisms, there are induced actions on the space of local functionals and consequently on the corresponding sh-Lie algebra. We determine conditions under which the sh-Lie structure induces an sh-Lie structure on a corresponding reduced space where the reduction is determined by the action of the group. These results are not directly a consequence of the corresponding theorems on Poisson manifolds as none of the algebraic structures are Poisson algebras.

Abstract.This paper is concerned with the development of a (discrete) groupvalued functor Ext defined on S£xSÛ where ¿? is the category of locally compact abelian groups such that, for A and B groups in ¿£, Ext (A, B) is the group of all extensions of B by A. Topological versions of homological lemmas are proven to facilitate the proof of the existence of such a functor. Various properties of Ext are obtained which include the usual long exact sequence which connects Horn to Ext. Along the way some applications are obtained one of which yields a slight improvement of one of the Noether isomorphism theorems. Also the injectives and projectives of the category of locally compact abelian totally disconnected groups are obtained. They are found to be necessarily discrete and hence are the same as the injectives and projectives of the category of discrete abelian groups. Finally we obtain the structure of those connected groups C of -S? which are direct summands of every G in ä? which contains C as a component.This paper is concerned with homological algebra in the category of locally compact abelian groups and with certain of its applications. In the first part of the paper we lay down the necessary homological framework for the work which is to follow. This framework provides for the development of a functor Ext which generalizes the usual functor Ext as is defined in (discrete) abelian group theory and also generalizes the functor Ext discussed by Moskowitz [10]. We show that Ext has many of the properties as does the ordinary Ext functor whose arguments are discrete groups.The second part of the paper (roughly from §3 on) deals with some applications. In particular, we completely determine the structure of those connected locally compact abelian groups G having the property that whenever H is locally compact abelian and H has G as its component, then G is a direct summand of //. We actually prove a much better theorem (Theorem 5.2). Other theorems of interest are obtained along the way. For example, the injectives and projectives of the