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# Homology, Homotopy and Applications

## Volume 13 (2011)

### Number 1

### Cyclic structures in algebraic (co)homology theories

Pages: 297 – 318

DOI: http://dx.doi.org/10.4310/HHA.2011.v13.n1.a12

#### Authors

#### Abstract

This note discusses the cyclic cohomology of a left Hopf algebroid ($\times_A$-Hopf algebra) with coefficients in a right module-left comodule, defined using a straightforward generalisation of the original operators given by Connes and Moscovici for Hopf algebras. Lie-Rinehart homology is a special case of this theory. A generalisation of cyclic duality that makes sense for arbitrary para-cyclic objects yields a dual homology theory. The twisted cyclic homology of an associative algebra provides an example of this dual theory that uses coefficients that are not necessarily stable anti Yetter-Drinfel'd modules

#### Keywords

cyclic homology; Hopf algebroid; twisted cyclic homology; Lie-Rinehart algebra

#### 2010 Mathematics Subject Classification

16E40, 16T05, 16T15, 19D55, 58B34