On the q-analog of homological algebra

TL;DR

This paper generalizes homological algebra to N-complexes where the differential satisfies d^N=0, introducing q-deformations and a new homology theory involving trigonometric sums, with applications to differential forms and curvature.

Contribution

It introduces the concept of N-complexes with differentials satisfying d^N=0 and develops a corresponding homology theory using q-deformations, extending classical homological algebra.

Findings

Homology of N-complexes forms (N-1)-complexes that are (N-1)-exact.

A q-deformed theory of differential forms with covariant formalism and N-form curvature.

For N=3, the curvature resembles the Chern-Simons functional.

Abstract

This is an attempt to generalize some basic facts of homological algebra to the case of "complexes" in which the differential satisfies the condition $d^N=0$ instead of the usual $d^2=0$. Instead of familiar sign factors, the constructions related to such "N-complexes" involve powers of q where q is a primitive Nth root of 1. We show that the homology (in a natural sense) of an N-complex is an $(N-1)$-complex which is $(N-1)$-exact, and the role of the Euler characteristic is played by the trigonometric sum $\sum q^i \dim(C^i)$. By q-deforming the de Rham differential we develop a version of the theory of differential forms which is coordinate-dependent but covariant with respect to a natural Hopf algebra. In particular, there is a meaningful formalism of connections with the curvature being an N-form given by the N th power of the covariant derivative. For $N=3$ the expression for the curvature is very similar to the Chern-Simons functional. This text was written in 1991.