nLab
N-complex

NN-complexes

Idea

While (co)chain complexes involve d 2=0d^2 = 0, (co)chain NN-complexes involve a differential dd satisfying d N=0d^N = 0, where NN is a positive integer bigger than 22.

References

The N=3N=3 case has been studied in

  • W. Mayer, A new homology theory, I, II, Annals of Math. 43 (1942), 370–380, 594–605.

The modern attention to the subject started in

  • M. M. Kapranov, On the q-analog of homological algebra, q-alg/9611005

  • Michel Dubois-Violette, Lectures on differentials, generalized differentials and on some examples related to theoretical physics, math.QA/0005256

  • Michel Dubois-Violette, Tensor product of N-complexes and generalization of graded differential algebras, Bulg. J. Phys. 36 (2009) 227–236 pdf

  • Michel Dubois-Violette, Marc Henneaux, Tensor fields of mixed Young symmetry type and N-complexes, math.QA/0110088 doi

In the following article the NN-homological versions of Tor and Ext functors are expressed in terms of classical Tor and Ext (of course, with shifted indices). As a consequence of this type of result, the interest in the NN-homological algebra somewhat diminished.

  • Christian Kassel, Marc Wambst, Algèbre homologique des N-complexes et homologie de Hochschild aux racines de l’unité, Publ. RIMS, Kyoto Univ. 34 (1998), p. 91-114 pdf q-alg/9705001
category: algebra

Last revised on March 7, 2013 at 01:43:19. See the history of this page for a list of all contributions to it.