There is no question that the cohomology of infiniteยญ dimensional Lie algebras deserves a brief and separate monoยญ graph. This subject is not cover̃d by any of the traditionยญ al branches of mathematics and is characterized by relativeยญ ly elementary proofs and varied application. Moreover, the subject matter is widely scattered in various research papers or exists only in verbal form. The theory of infinite-dimensional Lie algebras differs markedly from the theory of finite-dimensional Lie algebras in that the latter possesses powerful classification theoยญ rems, which usually allow one to "recognize" any finiteยญ dimensional Lie algebra (over the field of complex or real numbers), i.e., find it in some list. There are classificaยญ tion theorems in the theory of infinite-dimensional Lie alยญ gebras as well, but they are encumbered by strong restricยญ tions of a technical character. These theorems are useful mainly because they yield a considerable supply of interestยญ ing examples. We begin with a list of such examples, and further direct our main efforts to their study
CONTENT
1. General Theory -- ยง1. Lie algebras -- ยง2. Modules -- ยง3. Cohomology and homology -- ยง4. Principal algebraic interpretations of cohomology -- ยง5. Main computational methods -- ยง6. Lie superalgebras -- 2. Computations -- ยง1. Computations for finite-dimensional Lie algebras -- ยง2. Computations for Lie algebras of formal vector fields. General results -- ยง3. Computations for Lie algebras of formal vector fields on the line -- ยง4. Computations for Lie algebras of smooth vector fields -- ยง5. Computations for current algebras -- ยง6. Computations for Lie superalgebras -- 3. Applications -- ยง1. Characteristic classes of foliations -- ยง2. Combinatorial identities -- ยง3. Invariant differential operators -- ยง4. Cohomology of Lie algebras and cohomology of Lie groups -- ยง5. Cohomology operations in cobordism theory. -- References
