Cohomology of Infinite-Dimensional Lie Algebras

Description

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. Contents Chapter 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 Chapter 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 Chapter 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 8 21 30 40 54 65 77 89 Appendix 1. Cohomology of Lie Algebras of Generalized Jacobian Matrices 2. Stable Cohomology of the Algebra Wn with Tensor Coefficients and Generators and Relations for the Algebra L1(n) 3. Invariant Differential Operators in General Position on the Line (see 3.3.3) REFERENCES 5