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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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