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206 BOOK REVIEWS Given what has been said above, it suffices to construct a resolution D ~, D 2 .... for the differential operator D. This can be done now purely algebraically, using the super Lie algebra structure on the space of all G-valued differential forms on G: D1(6) = d~ + [8, to] for ~ a G-valued one-form on G. The succeeding operators D 2, D 3 .... are defined similarly. The above formulas are closely linked to Lie algebra cohomology in the Chevalley-Eilenberg sense. When the relations between deformation theory in the Kodaira-Spencer sense and the cohomology are worked out in detail, one obtains the cohomological approach to Lie algebra deformation theory developed by Nijenhuis and Richardson. Now, if G is an infinite-dimensional Lie algebra of vector fields, the si*uation is more complicated. Spencer's great achievement was to construct a formalism which works in case G generates a locally transitive pseudogroup on X and is defined by a system of linear partial differential equations. Alternately, one can think of Cartan's work as constructing a substitute for the space 'G' used (as sketched above) in the finite-dimensional case, in terms of a projective limit of finite-dimensional manifolds. One
Acta Applicandae Mathematicae – Springer Journals
Published: May 5, 2004
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