Cohomology of Infinite-Dimensional Lie Algebras

by D.B. Fuks

Overview

There is no question that the cohomology of infinite­ dimensional Lie algebras deserves a brief and separate monograph. 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 classification 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 interesting examples. We begin with a list of such examples, and further direct our main efforts to their study.

Product Details

ISBN-13:	9781468487671
Publisher:	Springer US
Publication date:	06/12/2012
Series:	Monographs in Contemporary Mathematics
Edition description:	1986
Pages:	339
Product dimensions:	5.98(w) x 9.02(h) x 0.03(d)

Table of Contents

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.