Lie algebra cohomology and index theory

Time: Monday, 13-15 HG E22
First meeting: Monday, 30 October
Exercises: The course will be complemented by an exercise class run by Carlo Rossi. The time will be fixed at the first meeting.
Summary: In 1989 Feigin and Tsygan proposed an algebraic approach to the Riemann-Roch-Hirzebruch (RRH) formula, based on the Hochschild homology of the algebra of polynomial differential operators. In particular they constructed a distinguished class in the cohomology of the Lie algebra of polynomial vector fields mapping via formal differential geometry to the RRH characteristic class. Recent progress in this subject is based on new developments in deformation quantization, in particular the Tsygan formality conjecture proved by Shoikhet. After introducing basic notions of Lie algebras and their cohomology, we will review this theory and discuss some recent applications to traces in deformation quantization and to extensions of the RRH theorem.
Tentative Programme: Lie algebras, derivations, extensions—cohomology of Lie algebras with first applications and relation to differential geometry—The cohomology ring of gl_N—The Lie algebra of formal vector fields and its cohomology—Hochschild homology of the algebra of differential operators—Formal differential geometry—The Riemann-Roch-Hirzebruch formula.
Prerequisites: the first part of this course will be elementary and uses basic multilinear algebra. Then we will use basic notions of differential geometry and de Rham cohomology. In the last part of the course it will be useful to know some basic complex algebraic geometry, as can be found in Chapter 0 of Griffiths-Harris.
Literature: Feigin, B. L.; Tsygan, B. L., Riemann-Roch theorem and Lie algebra cohomology. I. Proceedings of the Winter School on Geometry and Physics (Srni, 1988). Rend. Circ. Mat. Palermo (2) Suppl. No. 21 (1989), 15-52.
Fuks, D. B., Cohomology of infinite-dimensional Lie algebras. Contemporary Soviet Mathematics. Consultants Bureau, New York, 1986. xii+339 pp.
Loday, Jean-Louis, Cyclic homology. Grundlehren der Mathematischen Wissenschaften, 301. Springer-Verlag, Berlin, 1998. xx+513 pp.
I. N. Bernsteın and B. I. Rosenfeld, Homogeneous spaces of infinite dimensional
Lie algebras and the characteristic classes of foliations, Uspehi
Mat. Nauk 28 (1973), no. 4(172), 103–138 (Russian); English transl., Russian
Math. Surveys 28 (1973), no. 4, 107–142. MR 0415633 (54 #3714)