The abstract structure of the Batalin-Vilkovisky formalism is nicely captured in operadic language, so that 1-shifted Poisson algebras encode the algebra of classical theories and BD algebras encode the algebra of quantum theories. We explain this language and then discuss what it means for a Lie algebra (or higher versions) to act on such algebras. Our primary aim is then to extract consequences at the level of factorization algebras for field theories. A version of Noether's first theorem appears at the classical level, and its BV quantization recovers Ward-Takahashi identities, all reinterpreted as maps of factorization algebras. We also hope to discuss examples from chiral conformal field theory, such as the action of the Virasoro algebra on well-known chiral algebras.

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