# Cyclic homology

`2019-11-20 10:30:00``2019-11-20 11:30:00``Cyclic homology``For an algebra A over a unitary commutative ring k, we have the Hochschild homology HH_*(A). One use of it was a generalization of the modules of differential forms to non-commutative algebras. This gave us HDR_*(A), the non-commutative de Rham homology, developed by Alain Connes in his paper “Non-commutative differential geometry.” In that paper he also produced cyclic homology, HC_*(A), which is connected to both Hochschild and de Rham homology. The nicest connection between them is when k contains Q. Then we get the Karoubi exact sequence 0 —> HDR_n(A) —> HC_n(A) —> HH_{n+1}(A). In the study of quantum groups, cyclic homology is generalized to twisted cyclic homology for a pair of an algebra together with a given k-algebra automorphism. I was able to extend Karoubi’s theorem to twisted cyclic homology and also to twisted cyclic homology for crossed product algebras (an algebra together with a group of k-algebra automorphisms). Another extension of cyclic homology is to coalgebras, producing a cyclic cohomology. One example would be the coalgebra of a Frobenius algebra.``Third floor seminar room (room 201, building 216)``אוניברסיטת בר-אילן - Department of Mathematics``mathoffice@math.biu.ac.il``Asia/Jerusalem``public`For an algebra A over a unitary commutative ring k, we have the Hochschild homology HH_*(A). One use of it was a generalization of the modules of differential forms to non-commutative algebras. This gave us HDR_*(A), the non-commutative de Rham homology, developed by Alain Connes in his paper “Non-commutative differential geometry.” In that paper he also produced cyclic homology, HC_*(A), which is connected to both Hochschild and de Rham homology. The nicest connection between them is when k contains Q. Then we get the Karoubi exact sequence

0 —> HDR_n(A) —> HC_n(A) —> HH_{n+1}(A).

In the study of quantum groups, cyclic homology is generalized to twisted cyclic homology for a pair of an algebra together with a given k-algebra automorphism. I was able to extend Karoubi’s theorem to twisted cyclic homology and also to twisted cyclic homology for crossed product algebras (an algebra together with a group of k-algebra automorphisms). Another extension of cyclic homology is to coalgebras, producing a cyclic cohomology. One example would be the coalgebra of a Frobenius algebra.

Last Updated Date : 06/11/2019