1996/11/04 by Mikhail Kapranov, Kapranov, M. M. · 2 citations
Mathematics · #Advanced Topics in Algebra #Algebraic structures and combinatorial models #FOS: Mathematics #Homotopy and Cohomology in Algebraic Topology #Quantum Algebra (math.QA)
paper · pdf · doi:10.48550/arxiv.q-alg/9611005
openalex publication_date 1996/11/04 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
This is an attempt to generalize some basic facts of homological algebra to the case of "complexes" in which the differential satisfies the condition dN=0 instead of the usual d2=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 ∑ qi dim(Ci). 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.