This book has three related but distinct parts. In the first part of the book, we construct a new sequence of generators of the BRST complex and reformulate the BRST differential so that it acts on elements of the complex much like the Maurer-Cartan differential acts on left- invariant forms. In particular, for an important class of physical theories, we show that in fact the differential is a Chevalley-Eilenberg differential. In the second part of the book, we isolate a new concept which we call the chain extension of a $D$- algebra. We demonstrate that this idea is central to to a number of applications to algebra and physics. Chain extensions may be regarded as generalizations of ordinary algebraic extensions of Lie algebras. Applications of our theory provide a new constructive approach to BRST theories which only contains three terms. Finally, we show that a similar development provides a method by which Lie algebra deformations may be encoded into the structure maps of an sh-Lie algebra with three terms.

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