Cohomology of Infinite-Dimensional Lie Algebras

ISBN-13: 9781468487671 / Angielski / Miękka / 2012 / 339 str.

D. B. Fuks

Opis

There is no question that the cohomology of infinite- dimensional Lie algebras deserves a brief and separate mono- graph. This subject is not cover d by any of the tradition- al branches of mathematics and is characterized by relative- 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 theo- rems, which usually allow one to "recognize" any finite- dimensional Lie algebra (over the field of complex or real numbers), i.e., find it in some list. There are classifica- tion theorems in the theory of infinite-dimensional Lie al- gebras as well, but they are encumbered by strong restric- tions of a technical character. These theorems are useful mainly because they yield a considerable supply of interest- ing examples. We begin with a list of such examples, and further direct our main efforts to their study.

Spis treści

1. General Theory.- §1. Lie algebras.- §2. Modules.- §3. Cohomology and homology.- §4. Principal algebraic interpretations of cohomology.- §5. Main computational methods.- §6. Lie superalgebras.- 2. Computations.- §1. Computations for finite-dimensional Lie algebras.- §2. Computations for Lie algebras of formal vector fields. General results.- §3. Computations for Lie algebras of formal vector fields on the line.- §4. Computations for Lie algebras of smooth vector fields.- §5. Computations for current algebras.- §6. Computations for Lie superalgebras.- 3. Applications.- §1. Characteristic classes of foliations.- §2. Combinatorial identities.- §3. Invariant differential operators.- §4. Cohomology of Lie algebras and cohomology of Lie groups.- §5. Cohomology operations in cobordism theory..- References.