2006/07, 5th semester

Course Materials
Cyclic Homology Theory

This course on cyclic homology is split into 3 parts. The first and the third one will be taught by Jean-Louis Loday, and the second one by Mariusz Wodzicki. Part I is devoted to the very foundations. Part II reveals connections between advanced homological calculations regarding pseudodifferential symbols and operators, noncommutative residue and the Virasoro cocycle. Part III deals with topics ranging from Lie algebras to graph-complexes. As a prerequisite for this course, the knowledge of basic notions of homological algebra is expected, and the knowledge of very basic differential geometry (familiarity with differential forms and connections) is encouraged.

Part I:

One way of discovering the cyclic (co)homology theory is to unravel completely the most simple simplicial model of the circle. Comparing its simplicial structure with the n-stratum (identified with the cyclic group of order n+1) gives right away the construction of the cyclic category. Then follows the definition of a cyclic module and all its (co)homology theories: simplicial, cyclic, negative cyclic, periodic. Then, one can apply them to different kinds of cyclic modules: among the most important ones are those which come from an associative algebra and those which come from a Hopf algebra. On the topological side, cyclic spaces give rise to topological spaces equipped with an action of the circle. The Chern character, which, historically, was the motivation for Alain Connes to set up the cyclic cohomology theory, permits us to compare algebraic K-theory to cyclic homology.

Part II:

This part is concerned with some more advanced computations. A detailed calculation of symbols, and of the algebra of pseudodifferential operators (of unbounded order as well as of order zero) will be presented. Connections with the noncommutative residue, the higher noncommutative residue, and the Virasoro cocycle will be described.

Part III:

An important relationship with Lie algebras is given by the computation of the Lie algebra homology of matrices in terms of cyclic homology (Loday-Quillen-Tsygan theorem). A variation involving Hochschild homology needs the notion of Leibniz algebras. It turns out that Leibniz algebras are strongly related to the Schouten-Nijenhuis bracket. The same technique applied to Lie algebras of derivations of algebraic operads leads to different kinds of graph-complexes (a la Kontsevich). Thus a graph-complex can be interpreted as an operadic version of the cyclic complex. Their importance is due to the relationship with Teichmuller spaces and the outer automorphisms group of a free group. The Leibniz version leads to non-commutative graph-complexes, yet to be fully explored.

Course Summary:

  1. Simplicial objects, cyclic objects.
  2. Hochschild and cyclic homology of algebras (co)acted by Hopf algebras (standard and Hopf-cyclic homology groups)
  3. Computation of cyclic homology groups.
  4. S1-spaces, algebraic K-theory and the Chern character.
  5. Homology of the algebras of pseudodifferential symbols and operators.
  6. Coinvariant theory.
  7. Algebraic operad.
  8. Lie and Leibniz homology of derivation Lie algebras.
  9. Graph-complexes and their non-commutative analogues.

Part I
Jean-Louis Loday
Notes by P. Witkowski
Download Last update
Table of contents
  1. Cyclic category
    1. Circle and disk as a cell complexes
    2. Simplicial sets
    3. Fibrations
    4. Cyclic category
    5. Noncommutative sets
    6. Adjoint functors
    7. Generic example of a simplicial set
    8. Simplicial modules
    9. Bicomplexes
    10. Spectral sequences
  2. Cyclic homology
    1. The cyclic bicomplex
    2. Characteristic 0 case
    3. Computations
    4. Periodic and negative cyclic homology
    5. Harrison homology
    6. Derived functors
  3. Relation with K-theory
    1. K-theory
    2. Trace map
    3. Algebraic K-theory
Part II
Mariusz Wodzicki
Notes by P. Witkowski
Download Last update
Table of contents
  1. The algebra of symbols
    1. Local definition of the algebra of symbols
    2. Classical pseudodifferentials operators
    3. Statement of results
    4. Derivations of the de Rham algebra
    5. Koszul-Chevalley complex
    6. A relation between Hochschild and Lie algebra homology
    7. Poisson trace
    8. Hochschild homology
    9. Cyclic homology
      1. Further analysis of spectral sequence
      2. Higher differentials
  2. Appendix A: Topological tensor products
  3. Appendix B: Spectral sequences
Part III
Jean-Louis Loday
Notes by P. Witkowski
Download Last update
Table of contents
  1. Homology of Lie algebras of matrices
    1. Leibniz algebras
    2. Computation of Lie algebra homology H_*(gl(A))
    3. Computation of Leibniz homology HL_*(gl(A))
  2. Algebraic operads
    1. Schur functors and operads
    2. Free operads
    3. Operadic ideals
    4. Examples
    5. Koszul duality of algebras
    6. Bar and cobar constructions
    7. Bialgebras and props
    8. Graph complexes
    9. Symplectic Lie algebra of the commutative operad
  3. Appendix A: Cyclic duality and Hopf cyclic homology
    1. Cyclic duality
    2. Cyclic homology of algebra extensions
    3. Hopf-Galois extensions
    4. Hopf cyclic homology with coefficients
  4. Appendix B: Twisted homology and Koszul
Exam Exam questions

The exam was on 30th January 2007. It consisted of the written part (five exercises) and oral part. In the oral part each student had to answer two questions: easy one and difficult one (chosen from the two difficult questions).

Four students (on the graduate and undergraduate level) passed the exam.

Exam, written and oral

Designed by: Pawel Witkowski